PINAKI SANKAR RAY, M.Sc. (CALCUTTA)

Home , Bethe formula

THEORY OF ABSORFHON OF HIGH ENERGY (MeV-GeV) NUCLEI AND OF REHEAT FOR THERMONUCLEAR REACTIONS IN INERTIALLY CONFINED LASER COMPRESSED PLASMA

THESIS SUBMITTED FOR THE DEGREE OF

DOCTOR of PHILOSOPHY

IN THE FACULTY OF SCIENCE

THE UNIVERSITY OF NEW SOUTH WALES

AUGUST 1977 y 'IVERSITY OF N.S.W.

43303 2VAPR.78 LIBRARY riiwy1^ mm'**** (i)

ACKNOWLEDGEMENT

The author is deeply indebted to Professor Heinrich

Hora, Head of the Department of Theoretical Physics,

for suggesting the problem and supervising the research. He has introduced him to the subject of

laser fusion, and above all his enthusiasm for work has been the author's greatest encouragement. (ii)

ABSTRACT

The theory of laser induced nuclear fusion reactions under inertial confinement has been investigated in this work. The special reactions treated numerically are the deuterium-tritium and the hydrogen-boron(11) fusion in plasma of densities corresponding to the solid state and higher. The energy gain has been calculated both with and without the incorporation of plasma heating due to the absorption of the MeV alphas released in the reactions. This reheat by the alphas is related to their range and the concept of the penetration length of energetic charged particles in plasma has been discussed from the point of view of the Fokker-Planck formalism. As a point of theoretical interest the process of soft photon emission during scattering as necessitated by quantum electrodynamics has also been incorporated in the Fokker-Planck method.

In this case one does not need the Coulomb logarithm to avoid the divergence of small angle scattering. However there arises another term due to quantum electrodynamic cut-off. A new model called the "collective model" has been constructed for the calculation of the range of high energy charged particles in high density plasma. This (iii) model is very close to the Bethe-Bloch type theory in use

in nuclear physics and Bagge's modification of it for the penetration of high energy electrons in plasma. This

calculation differs essentially from the Fokker-Planck

theory for high density hot plasma — it predicts range

for particles of MeV initial energy shorter by a factor of 1000 in some cases. Furthermore the range decreases with the rise of plasma temperature in contrast to the

Fokker-Planck calculations. It is also suggested that

the collective model should be applied for the study of heavy-ion injection in plasma and some numerical results of uranium range are illustrated.

For the calculation of fusion efficiency the

cooling of the plasma due to adiabatic expansion has been

taken into account. The break even energies of DT and

HB11 for solid state density are found to be 1.5x106 J

and 2.5xl013 J respectively without the incorporation of

reheat. Then it has been shown how to introduce the

reheat mechanism together with ionic depletion. In this

case the equations governing the thermokinetic expansion

can only be treated numerically due to complexity. The

influence of reheat is noted only at higher densities —

for the solid state density it is negligible. One

significant aspect of this addition of reheat is the

ignition. This can be defined as the optimised stage where, although the initial temperature of the fuel is

relatively low, the absorption of hot alphas produced by

the few initial reactions is strong enough to offset the (iv)

temperature drop due to inertial expansion so that the temperature gets raised to a point where the reaction cross

-section is large. This occurs only at high densities, e.g., for DT plasma of initial density 103 times the solid state and initial temperature 2 keV with initial volume

10 cm3 one would obtain a gain of 210 for input laser energy of 6 kJ. For HB11 plasma with initial density 105 times the solid state density — which is theoretically possible under the concept of nonlinear force mechanism as developed by Hora — and temperature 22 keV and volume

10 -8 cm ^ a gain of 22 is reached for input energy of 2 MJ. CONTENTS

Acknowledgement (i)

Ab s t r a c t (ii)

List of Publications (v)

Chapter I Introduction 1

Chapter II Resume of cross-sections of

nuclear reactions 6

Chap ter III Models of the range of energetic

charged heavy particles in plasma 19

3.1 The Fokker-Planck formalism 20

3.2 Energy loss in Fokker-Planck formalism 27

3.3 Coulomb Interaction 31

3.4 Scattering with soft photon emission 36

3.5 The collective model for energy loss 44

3.6 Relation to the Bethe-Bloch formalism

and Bagge's modification ^9

3.7 Numerical discussions and conclusions 55

Chapter IV Introduction 64

4.1 Model of a high density expanding plasma 65

4.2 Mathematical formulation of the Thermo-

kinetic expansion theory 67

4.3 .Fusion yield in adiabatica1ly expanding

plasma ^0

4.4 Contribution from alpha particle reheat 73

4.5 Numerical results and discussions 80 Chapter V Heavy Ion Penetration 101

Conclusion 109

Appendix I 1 1 1

Appendix II 113

Appendix III 115

Ref erences 119

Computer Program 125

Photocopies of Publications 129 (v)

LIST OF PUBLICATIONS

H. Hora and P.S. Ray: Bull. Am. Phys. Soc. , 2_1, 73 , (1976)

"Reaction Gain of Hydrogen-Boron (11) at Laser

Compressed Plasmas".

P.S. Ray and H. Hora: Nuclear Fusion, 16 , No.3 , 535-36 ( 1976)

"On the range of alpha particles in laser produced

superdense plasma".

P.S. Ray and H. Hora: At omkernenergie, 2j8, No. 3 , 15 5-157 ( 1976)

"Penetration Length of Alpha particles at Laser-

produced Thermonuclear reactions".

P.S. Ray and H. Hora: Bull. Am. Phys. Soc., 21, 1153 (1976)

"Thermalization Length of Alphas and other Nuclear

particles in High Temperature Plasmas".

P.S. Ray and H. Hora: Zeit. f. Naturforschung, 3 2A,

538-543 (1977)

"On the Thermalisation of Energetic charged particles

in Fusion plasma with Quantum E1ectrodynamic

considerations".

R. Castillo et al: Nuclear Methods and Instruments,

144 , 2 7-32 , (197 7)

"Advanced Fuel Nuclear Reaction Feasibility Using

Laser Compression II".

P.S. Ray and H. Hora: Laser Interaction and Related Plasma

Phenomena, eds. H.J. Schwarz and H. Hora, Vol.4B,

p 1081-1102, Plenum (N.Y.), 1977

"Corrected penetration Length of Alphas for Reheat

Calculations".

H. Hora and P.S. Ray: Atomkernenergie, 30 , 261 (197 7 )

"Ueber die elektrostatische Bindungsenergie von Plasma

und ihr Zusammenhang mit E1ementarkonstanten". 1

CHAPTER I

INTRODUCTION:

This thesis encompasses investigations on the

theory of energy release by nuclear fusion and some problems

related to it. Fusion denotes nuclear reaction processes

in which lighter nuclei combine under nuclear forces to

form heavier nuclei but where the mass differences involved

imply the release of enormous energy. Under the present

understanding of physics there exists in nature four

categories of forces or interactions, namely gravitation, weak interactions, electromagnetic and strong nuclear

forces. The last mentioned is the least understood of

these at least from a theoretical point of view although

in recent years there has been a great accumulation of

experimental results. The nuclear forces are of very short

range and are attractive at first then becoming repulsive

at very short distances. The attractive part is respon

sible for the binding of the nuclei of elements which are

constituted of protons and neutrons. The strength of

these forces are much greater than the electromagnetic

forces whereas the range as mentioned earlier much less 2

e.g. almost zero for atomic distances. Thus in order that fusion should take place the respective nuclei should be close enough. In the normal state of matter this is prevented by the repulsive electric forces between different nuclei due to their positive electric charges. One of the main problems for a successful fusion reaction to perform is therefore that of overcoming this repulsion. Now every nucleus is characterised by two fundamental integers A the mass number and Z the number of protons in it which defines its charge apart from other quantum numbers as spin etc.

If N be the number of neutrons in the nucleus one has

A = Z + N. For stable nuclei Z - N - . Thus the heavier the nuclei the electric repulsion between them is higher. One has therefore almost always considered the feasibility of fusion with different isotopes of hydrogen for example the deuterium-deuterium^ reaction which proceeds in two ways

D + D -+ He3 + n

D + D -* T3 + p

Although fusion reactions can be achieved in the laboratory by bombarding a target with energetic nuclear projectiles obtained with the help of accelerators one cannot realise a sustained reaction in this way which is necessary for successful energy production - in a cold target the electrons will absorb most of the energy of the incident projectiles. The only method available at present for this is to raise the temperature of the nuclear fuel so 3

high that the random thermal motion enable the nuclei to overcome the Coulomb repulsion and thus initiate fusion.

At this high temperature matter is always in the plasma state and this is the main reason for the importance of the knowledge of plasma physics for thermonuclear reactions.

In this work we shall be concerned with fusion reactions solely from the view point of plasma physics.

One concept which has been long in vogue is to attain thermonuclear reactions for plasma of relatively low density confined by magnetic field into toroidal or mirror configuration. A recent progress in this direction has 2 ) been the construction of tokamak for this purpose

However, in recent years the invention of laser beams of high power has been a significant development following an entirely different principle where the nuclear fuel initially in solid state is plasmatized by irradiating with pulsed laser radially from all directions 3)

Due to gasdynamic ablation it is conjectured that one can reach very high density and temperature which should be sufficient for the achievement of thermonuclear reactions.

This way of achieving fusion is called "inertial confinement" technique.

A very important contribution for the fusion plasma heating is that of the absorption of the energetic nuclei of the reaction products within the plasma itself which should favour or even sustain the reaction. This depends on the range of these particles. We have studied 4

this aspect in detail in Chapter III. It seems to us

that the result depends on the special model adopted and in

the absence of direct experimental evidence it is difficult

to decide for the correct view. However, we shall give 4 ) some arguments in favour of a "collective model" which closely follows the Bethe-Bloch type calculations of nuclear physics. It seems that in plasma physics one has not considered this one to date.

We have also studied one exotic reaction, e.g.,

the hydrogen-boron fusion^. This has a special advantage of no neutron emission so that it could be completely

"clean" from any hazard - this reaction produces charged alphas only. Furthermore, in this case electrical energy can be obtained directly from the alphas without the need of a thermal cycle of conversion.

We now give the content of the thesis. In

Chapter II we present a general review of nuclear reactions relevant for our purpose. In Chapter III we present the

Fokker-Planck formalism widely in use in plasma physics and describe the calculation of the range of charged heavy particles with high initial energy. We then construct the collective model to be used for high density plasma.

In Chapter IV we have shown how to calculate

the reaction efficiency for inertial confinement. Calcula

tions have also been done with the incorporation of alpha particle reheat. We have defined an "optimum temperature" 5

for a fuel from the numerical results obtained. This may have some implications for the design of the laser-target systern.

In Chapter V we present some numerical results 9 2 of the penetration depth of heavy ions, e.g., U„ 0 for high Z jo density plasma. This estimate is of importance for fusion by heavy ion bombardment and plasma heating which some people have suggested recently. 6

CHAPTER II

RESUME OF CROSS-SECTIONS OF NUCLEAR REACTIONS:

In principle ,nuclear reactions take place when nuclei are in collision. The theory so far developed deals with two body collisions, e.g., nuclei of species X colliding with nuclei of species a. Out of such an interaction one obtains two or more nuclei or even one of the nuclei could be captured. Most of the fusion reactions considered so far has two nuclei in the final state, e.g. :

X + a -* Y + b (2.1)

where Y and b denote some nuclei. Such a reaction is also denoted as X (a,b) Y. The final state may not always be the same, i.e., one can have also

r Z + c

X + a -* (2.2)

l W + d 7

Under many conditions of collissions the nuclei emerging

out of the interaction are of same species as those entering

it and these are referred to as "scattering". These type

of processes are in most cases governed by electromagnetic

interactions of the nuclei involved. Although there does not exist an exact definition of "nuclear reaction", one usually classifies under this name those collisions processes where the species of nuclei undergo changes. They are

governed by nuclear forces. For these to take place the

energy of motion should be high enough to overcome the

electrostatic repulsion which is given by the "Coulomb barrier".

The barrier energy is defined in terms of the distance

of closest approach which is the sum of the radii of

the two nuclei in interaction:

Z i Z ^ e' E (2.3) c

whe r e Z ^ , specify the respective charges, The radius of 1/3 a nucleus is calculated with the formula R = where m c TT ■h is the Planck’s constant divided by 2tt, m mass of the

tt meson, c velocity of light and A the mass number.

Only when the kinetic energy of relative motion is higher

than E^ can the nuclei come close enough for a nuclear

reaction to take place.

For a reaction like (2.1) usually the rest mass of

the product is different from that of the initial nuclei

present. If it is higher energy must be supplied externally 8 in order that the process could take place. For the case where the rest mass of product is smaller than that initially present this difference would be released as energy of the reaction. We shall consider only such processes - the fusion reaction belong to this category. These are most explicitly denoted as

X + a + Y + b + Q where

Q = [ M (X ) + M ( a ) - M ( Y ) - M(b)]c2

Here M(X) etc. represent the rest masses of the respective nuclei. We shall not consider processes where gamma rays may be emitted so that Q would denote energy in the form of kinetic energy which would be shared between Y and b.

For the estimation of the total energy yield by fusion in a nuclear fuel one has to calculate the rate at which the reaction proceeds which is defined through the "cross- section" for the reaction. Theoretically it is defined with the help of transition probability between the initial and final states of the reaction as defined in the quantum mechanics of collision. In the following the definition of the reaction cross-section in relativistic invariant form as viewed in elementary particle physics is given.

Consider the interaction of two particles A and B with four momenta p^ and p^ respectively. Suppose out 9 of their interaction n particles are produced with four momenta p^, p^» •••> Pn* That is the process is of the type

pr a +P*b1 +P*11 +Po+v 2 ••• + *nP

For simplicity we shall neglect other quantum numbers of the particles such as spin etc. From the viewpoint of quantum mechanics the system is conceived as undergoing transition from an initial state denoted |i> which is formed by the two momenta p and p, to the final state |f> which is v a v b 1 again defined by the momenta variable p^, p..., Pn*

f) 1 Heisenberg originally introduced the concept of the S-matrix to characterise the transition. In principle one has to construct it out of the interaction of the wavefields whose quanta are the particles considered. The matrix element of

S between |i> and |f> corresponds to the quantum mechanical transition amplitude. Since energy and momenta are to be conserved by the interaction one can write the following expression for this matrix element as:

4 .4 f i + i ( 2tt ) 6 (Pa+Pb“ £ pi) N < f | T I i > ’ i= 1

is the Kronecker symbol 6_ . = 0 if 1 i> ^ f > , 1 67 f , 1 1

. = 1 if | i> = f > f ,i 1

6 (p +p. - E p.) is the Dirac's delta function in four a *b . . *i 1=1 dimensions expressing the energy-momentum conservation;

N is a normalising constant which cancels out in the final expression for cross-section; and the new entity T is called the "transition matrix".

In the theory of elementary particles one then shows that the cross-section G for the process which describes the transition probability per unit time is given by

n n n 6 (p +p £ P . ) < f T i > 3n - 4 a a (2tt) i= 1 i= 1 2/A ( s ,m m2)

Here is the energy of the ith particle in the final product of reaction corresponding to four momenta p^ and the integration is over the space part of p^. The variable s denotes the square of the total centre of mass energy in the initial state, i.e.

CM. 2 s + E b '

where E , E, are the energies of A and B respectively a b in the centre of mass frame. One also calls s a Mandelstam variable. The function A(x,y,z) is defined as

2 2 2 A(x,y,z) = x +y + z - 2xy - 2yz - 2zx

Thus to get G one has to calculate theoretically the matrix element < f | T | i>. This way of calculating the cross-section for different processes has remarkable success in quantum

electro dynamics and also in the theory of weak interactions which deals with the beta decay of nuclei. For such cases

the matrix T can be explicitly evaluated from the interaction.

However, for strong interaction physics which governs the

nuclear forces inspite of many attempts there does not exist

any consistent theory except in bits and pieces. One assumes

in nuclear physics only a more simplified picture as in non re1atiativistic scattering theory of quantum mechanics.

One concept which has received great attention for the

calculation of cross-section is the "compound nucleus" model

first proposed by Bohr . ^ ^ According to this idea nuclear

reactions proceed in two steps. Firstly, the colliding

nuclei form a "compound" nucleus a form of an excited bound

states and then as the second step this compound nucleus

decays according to its possible modes of decay thereby

giving rise to various "channels" of the reaction. This decay

is analogous to the light emission from an excited atomic

state in quantum mechanics. Thus the reaction X(a,b)Y would proceed in the form

X+a+C*+Y+b

where C* denotes an excited compound nucleus. In this model the lifetime of the state C* is long enough so that

the mode of formation is forgotten by the compound nucleus

excepting for such as the total energy, parity, spin, etc.

The decay is then entirely independent from its formation.

The cross-section is then assumed to be given by the expression 12

Wh G(a,b) = O —- c W wh ere

W = EW,b

Here W^ is decay probability for a particular channel b and sum is made over all possible channels to obtain W.

Furthermore, 0 is the cross-section for the formation of the c compound nucleus. At the present understanding of the problem the compound nucleus model gives a fairly satisfactory agreement with the experimental data in the low energy region.

We now discuss the quantity which is of our direct interest - the rate at which nuclear reactions take place in the plasma state. From the definition it follows that the rate R would be given by the cross-section multiplied by the relative flux so that one can write for a unit volume

* ~y ~y * . | 3-> 3 R = f f nx(v x) n2(v2) | v ^ - v 21 o(|v^-v2|) d Vj d v2

~y~ —y where n^(v^) and n2^V2^ denote the number of respective

—y —y nuclei present with velocity v^ and v2. Assuming the latter are given by the Maxwe11-Bo1tzmann distribution at

temperature T one can write

2 2 !^v ^+m2v 2 3/2 3/2 , 3 - , 3 - (——) ff d v d v, o e 2KT n 1n 2 v 2KT ^ 2KT V1 V2

where K is the Boltzmann constant, (m^ , m2) the masses 13

and (n^,n2) the densities of the respective nuclei. Now with new variables (v,V) defined as:

V1 V 2

mlvl+nl2v2 ml+m2 and with

n ^n2<(Jv>

one can write

m 3/2 m 3/2 MV2+pv2

< O v > (--7T=-) (-T-Tr’rr' ) // d v d V a(v)v e 2KT 2ttKT 2ttKT

2 JL JL E a(E) e m1 KT dE j— ^>Z K 1 3/2 mr. )2 —/Try (;1

Here y is the reduced mass

m!m2

m ^ +m 2 and

2 mlV

the bombarding energy of particle 1 for the reaction. Thus knowing o(E) either theoretically or experimentally one can calculate from above. At low energies one sometimes uses an approximate formula

0(E) = exp (——) E /E jni I..

TIVK

cm sec Fi^. rule's

Miixwrll

lnncfion

tiv

crnyrs

«»(

TKeV lor Ref incident

I>1 .9 * tot

dentrnm . \ i ,.

H ’

anil tMicrjjy THKKMONIVLKAR

in «*:l 100

ivncl kcY.

ion

400 400

HKA(TH>X 14

K.W' 15

DT Q - 17.690 McV / urn

inn

pB 1 ' i

Q - 8.682 i

MeV i

DHe° i

Q = 18.353 i

(VleV i

m u

T: - keV

Fig.2. Hydrogen-Boron (11) Reaction.

Ref.10. 16

where A and B are constants taken usually to fit the g ) experimental data. Thomson nas given values of for certain plasma temperature. For the reactions we are interested we shall use the values available from the graphs which are drawn with the help of experimental data.

We reproduce this in fig. 1 and fig.2, here. We note that 3 -1 has dimensions cm sec in cgs units.

We now mention some of the nuclear reactions^ ^ which are of importance for fusion energy. As already stated, the most studied reaction in this respect has been the deuterium-deuterium^ reaction. It has two branches, the neutron and proton branch,both proceeding almost at equal rates with respective Q values:

D + D -* He3 + n + 3.27 MeV

D+D+T+H+4.03 MeV

One point to be mentioned is that how the total energy Q released is shared among the product particles. This can be calculated by using the centre of mass frame where the total momentum is zero. For two particles in the final state of say, masses (m^jiT^) and velocities (v ^ , v £) one has

ni 1V 1 m2V2

This implies that if (K^jK^) be the respective kinetic energies one must have

K1+K2 - Q

from which (K^jK^) can be determined. For example, for the deuterium-tritium reaction

D + T + He4 + n + 17.58 MeV

4 the alpha particle He will carry 3.52 MeV in kinetic energy whereas the neutron n will have 14.06 MeV. Compared to the DD reaction this has much higher rate so that this could be one of the important fuel for practical use in possible fusion reactions.

One reaction which we shall consider is the hydrogen- boron reaction:"*^

1 1 1 4 B + H + 3He + 8.682 MeV

This reaction is entirely different in character from the other reactions in view of the fact that the final state

contains three particles. We shall assume for our calculations

that each alpha particle carries Q/3 kinetic energy, i.e.

2.894 MeV. One should mention that the three (or more) particles final state are very complex to study theoretically as well as experimentally. One distinguishes in this connection two ways how such a reaction will proceed namely, the sequential process^ viewed as

a+A+D*+b+B*

B* -* c + C

and the simultaneous process

a + A^-b + c + C

Depending on the mode, the energy distribution of the emitted particles vary. In fact the same reaction may proceed in both ways depending on the relative energy of motion. 19

CHAPTER III

MODELS OF THE RANGE OF ENERGETIC CHARGED HEAVY PARTICLES

IN PLASMA

The energy released in each fusion reaction is normally distributed as kinetic energy among the products, i.e., the alphas, protons, neutrons, etc., which come out.

This initial energy is of the order of MeV whereas the plasma temperature is in the keV range. Thus the energe tic heavy particles should move recti1inearly from the point where they are created in the plasma by the nuclear reaction until they lose energy by collisions mainly with the plasma electrons to the value of average thermal energy. This distance travelled defines the "range" of these particles. A knowledge of this is important for various aspects, e.g., whether such particles will leave the plasma or be absorbed will depend on the range - and this has relevance regarding the self-heating of the plasma to be considered later. We mention that since neutrons do not have electric charge they do not exchange energy with the electrons and the theoretical study for them has therefore to be based on entirely different considerations as for the charged ones. We shall not 1 3) take this in account here. Some people have also considered the probability of nuclear scattering and reactions of the fusion products which will be also neglected here. 20

In most treatment of the problem the plasma is

regarded as a gas so that the collision mechanism by which

the energy transfer takes place can be studied with the

usual Boltzmann equation. Since however this equation

cannot be solved exactly one has to employ an approximate

treatment which is the Fokker-Planck (F.P.) formalism.

We discuss this aspect first and later we shall consider

another model which may be called "collective model".

The last mentioned is very similar to the Bethe-Bloch

formalism widely used in nuclear physics. This has

also a close similarity with the calculations of Bohr

along classical lines.

3 . 1_____The Fokker-P 1 anck Formalism

The F.P. formalism in the kinetic theory of matter

can be looked upon from two different points of view - one

as an approximate expression for the collision integral

in the Boltzmann equation and the other being a stochastic

description of the rate of change of probability distribut

ion function in an assembly of particles where a sort of

averaging has been undertaken to describe the changes of

% velocity. The stochastic aspect corresponds to a Mark

off process and has been discussed in a well known work 14) of Chandrasekhar . Although it has not been mentioned by Chandrasekhar, it seems that his work is a mathematically

expanded version of a discussion by Born^^. In the 2 1

literature of plasma physics one very often refers to the

16) work of Rosenbluth et al in this context which is just

a mathematically explanatory version of Chandrasekhar's

work.

We consider a collection of particles of

several species which will be denoted by suffixes a, b, ...

etc. Thus the distribution function for species a will

be written as f ( r, v, t) . The Boltzmann equation

for f3 is

9fa 9 f' / 9 f' + v (3.1) 9 t V 91 9 v

wh e r e m^ is the mass of a particle of species

a, F denotes the external force on it, and

) is the collisional contribution to the rate ' Q.

of change of f In the F.P. formalism it has

the special mathematical form

3 fa \ (f + * IwTV ( 'c

(3.2)

Here (Av.)a and ( Av . Av.)a are called the i 1 J

first two F.P. coefficients. They are given by the relations

<6 v.)a = Z /d v' fb( v' ) {Av±}ab (3.3) 22

whe re {Av.}ab describes the average change per unit 1

time in v of species a by collision with species b summed over all scattering angles. One has the relation

n k {Av^} = /dfi Oab u Av^ (3.3a) where Cf ^ denotes the scattering cross-section for species a and b , u the relative velocity of collision, Av^ is the change in the ith

-f component of v by collision with v* of species b and d^ the solid angle of scattering. Similarly the second coefficient has the form

( Av^ Av.)a = £ fd v' f^(v') {Av^Av.} (3.4) J b J and

n k {Av. Av.} = /d^ a , u Av. Av. (3.4a) i J ab i j

These relations can be easily derived from the stochastic derivation of F.P. equation as given by 14) Chandrasekhar . For this purpose consider a

—y stochastic process where W(u,t) is the probability

—V distribution for a variable u at time t . In order t o obtain W(u, t + At) for a -time increment

At one introduces a transition probability. Let

^ (u; Au) be the transition probability that u undergoes an increment Au in the interval At 23

to random process. Then one can write

W(u, t + At) = / W(u - Au,t) 4>(u - Au ; Au)d(Au). (3.5)

This relation means that the probability that the variable has a certain value t + At is given by the probability that it has the value u - AuA at instant t and then undergoes an increment Au and this has to be summed over all possible Au .

The above equation is known as the Chapman-Kolmogoroff equation. Assuming Au to be small and expanding by Taylor’s series one then obtains

W ( u,t) + |^ At + 0 ( A t 2)

I/ JIT/ + N „ 3W A . 1v a2W /A N 2 , v 32W W(u,t) -E Au. + %£ ^ (Au.) + £ ~ Au.Au... . 9 u . l . ~ 2 ' i . 9 u . 9 u . i j J i i i9u. i < j i j

2 2 X ^(u; Au) -l 4^- Au.+ hi -Mf (Au.)2 + l —Au.Au.. i 3u. l i gu2 i . 3u.3u. 1 J i

. d(Au)

= W(u,t) - E -z-- Au. - E W 3-- (Au.) . 9u. i . 9u . i ii ii

2 2 + h E -Mr < (Au.)2) + E — (Au.Au.) (3.6) . ~ 2 l ... 9u.du. i j l 9ui i

2 2 + h l W -5-— ( (Au. ) 2) + l W -- ( Au.Au.) . ^ 2 i . . . 9u.9u. i j i du± i

Here one has utilised the probability normalising relation

f ip (u ; Au) d(Au) 1 and one uses the following notations:

(A u^) - /Au^ i|^(u;Au) d(Au)

( (Au^) = / (Au. ) ^ ijj (u ; A u) d(Au) (3.7)

( Au . Au .) =/Au . Au. 0 (u ; A u) d(Au) i J i J '

Now if one introduces the average changes per unit time deno ted ( ) :

< Au.) i < Au.) l a v At

( (Au.)Z> < (Au.) > (3.8) l a v At

< Au . Au .) (Au . Au.) ____1___3_ l j av At one obtains

_9_W „ 9 (V(4u.) ) + h E (W( Au . Au ) .(3.9') 91 L 9u. lav . . 9 u. 9 u l j av i, j i j

This is the F.P. equation. In order to relate it to the collisional part in rfoltzmann equation we identify W

3. with the distribution function f (r, v, t) . One can then write

9 f a - £ (fa ( Av.) ) + k E , ( f (Av.Av.) 91 . 9v. l av . . 9v.9v . i J a v ii i»J i J

(3.10)

Here then one has for (Av.) etc the expression i a v 25

(Av.)a = /Av. ipa(v; Av) d(Av) 1 av 1

a -+ -* ->■ where ip (v; Av) d(Av) denotes the transition

-y probability per unit time that v goes to v + Av

due to random collissions. To find an expression for

this we recall that in the kinematics of collision the

magnitude of the relative velocity before and after the

collision is the same so that the relative velocity

vector only changes direction which defines the scatter

ing angle. Now the scattering cross-section is defined

in quantum mechanics as the transition probability per

unit time divided by the relative flux of particles which is given by the relative velocity multiplied by

the particle density. One can therefore write

ipa(v; Av) d(Av) = £ /o , dfi u f^(v') dv' . a b b where O is the differential scattering cross- a b section between particles of species a and those of species b, df2 the solid angle of scattering, u

the magnitude of the relative velocity given by

■ -> -> | u = I v ’ - v |

b and f (vT) the number of particles of species b with velocity v', i.e., the distribution function

for species b . Thus one can write

< A v) £/dv ’ f\v-) f O , u Av df2 (3.11) a v ab b 26

Here = sin 0 d 0 d cf) where 0 is the scattering

angle and an azimuthal angle. Similarly one can

write

( A v . Av.)a = I /dv' f^(v') /a , u A v . Av. dfi. (3.12) i J b ab i j

This justifies the formula (3.3) and (3.A). To find

Av^ one has to consider the collisions more explicitly.

In the following we assume only two species of particles

present and one is a heavy particle:

m >> m, a b

Consider now the collision of two particles. Let

denote the relative velocity and V the velocity

o f the centre of mass. One has then

u v - V (3. 13) -> MV m v + m, v' a b whe re m + m. a b

From this one gets

a v v + fT u

so that the change in v is given by

a ^ Av ;— Au (3. 1A)

To calculateAu , introduce a local set of co-ordinates (e such that Let 2 ’ e3)

u ' be the relative velocity after collision. Then

if 0 is the scattering angle one can write considering

fig. 3 the following expression for Au :

| A u 2 u sin Ay

(3.15) and

aS |CNI

3. 2____Energy Loss in Fokker-Planck Formalism

To find the range of a heavy particle in plasma one has to calculate the rate at which energy loss takes place due to electronic collisions. For this purpose

let us write the F.P. equation taking into account the higher order terms of expansion in the form

3 fa 3_f^ F_i 9 t Vi 3v. m 9 v . l a i

(3. 16) where F . is the ith component of the external force l and the F.P. coefficient of order n . We have used the notation 28

(n) 3v. 9 v ...... 9 v . 1 121 1 o 1 n

ab ab . s ( n) a (v , v , • • • , v ) 1 2 n

(each 1,2,3)

This form of the equation was first utilised by Kramers^^

~y ~y Let now *P(v) denote any function of v . Its average ip with respect to the distribution f is then defined as ^ - / ip f dv -j- = —— / f d v

— / \p f dv n

dv dvl dv2 dv3

-y -y where n = n(r,t) = / f dv is the local density.

We shall denote the heavy particle by suffix H and the electron by suffix e . If one multiplies (3.6) -> -> by iKv) and integrates with respect to v one obtains

-(H) ■777(H) (H) — n (ipF. ) (nH f )+ 3^7(nH ♦V"5 m.T H dv. 1 1 H 1 (3. 17)

(- 1) r , n(m) / He rH N ,-»• + l — f ,"D a (m) f > dV • m = 1 29

Here averaging is taken with respect to the heavy particle He distribution function and a,(m) N denotes the F.P. coefficient of order m describing interaction of

electrons and heavy particle. Now integrating by parts

one can write

r , _ (m) , He rH. + z . N m r He /T_ (m) , * ..H J \fj D (a, . f ) dv (-1) fa . \ (D i^) f dv (m; (m)

(H) / He ~ (m) , (-° nH “(m) D f

In the absence of external force one therefore gets

(H) 1 He _ (xn) (n„ *(H>) + (n *v.(H)) = l aVD'v“‘> 9t H 8 x Hi L. m . (m) m= i

m He (m) , He , 9 ip where a D i[> = a (v. ,v. • , v . ) i 8 v. 9 v. . 9v 1 1 12 m l ^ 12

2 Now setting ^ ( v ) = ^ m Zv . one gets from (3.17) H . l l

since all derivatives higher than second order are zero

(H) (H) i 9 ( 2V“' . 9 ( 2 * 9t (nH mH V } + ^ 9^7 (nH mH V Vi (3.18)

(H) (H) nTT mT7 (a . v . + h b . . 6 . . ) H H l i ij ij where a. = a(v.) and b.. = a(v.,v.) are the ii ij i J first two F.P. coefficients. To pick a particular 30

particle with coordinate and velocity (r , v ) we set

f (r, v, t) = 6 (r - r (t)) 6 (v - v (t)) which gives

_ -> n = 6 (r - r (t) )

~y ^ V(v) = Ip (v )

One then obtains after integration with respect to r -> and v

m ( £ a . v . + ^ £ b . . ) (3.19) d t h . 1 i .ii l l

Up to this no approximation has been made. The range

R of the heavy particle will then be given by

R KT -(-r^)

(3.20) vHdE H

where Eq is the initial energy of the heavy particle and KT the plasma temperature. For further

calculations one needs to know explicitly the F.P.

coefficients a.'s and b..’s to which we come i ij now . 31

3.3. Coulomb Interaction

For the Coulomb scattering of an electron from a heavy particle of charge Z e the differential scat- H tering cross-section is given by

da 4 c o se c 6/2 (3.21) dft

where p is the reduced mass of the colliding

_1 l particles: + m y e o r y - m e and u the relative velocity, 0 the scattering angle. This formula is a source of difficulty due to the divergence at 0=0. The only way known at present to get over this is to introduce a cut-off

0 = 0 ^ for smaH scattering angles. In plasma one assumes that the electrons situated beyond the

Debye sphere about the heavy particle suffer only small angle scattering. Now if 0 denotes the deviation suffered by an electron at an impact parameter b from the heavy particle one has

. $ tan — m u b e

If one now sets b = A^ which should give small angle

3 $ = 0 . and sets ^ m u ~2 KT which is the average min e 32

thermal energy one obtains

3 3 2 3KT K T (3.22) 0 . 4 3 TIN mm 4ttN e ) 2e Z, V ( e we shall write A for e which is the factor for min

"Coulomb logarithm Utilising, (3.14) and (3.15) one can therefore write

72 4 - e ^ sin 0/2+(e2 cos (() le^sin (J))cos 0/2 dU ZH e u A v dft 2m mTT 2 . 3 e H u s m 0/2

c da u^ r77 cos 0/2 Then A v d$l d 0 (3.23) dft u u sin 0 / 2 . 0

u -*■ where one uses — = e, due to the special choice of u 1 axes. Now the integral on the right diverges at 0=0 which is a common problem whenever one uses Rutherford scattering cross-section. One uses in plasma physics always the cut-off defined in (3.22) so that one obtains

7T TT cos 0/2 cos 0/2 d 0 -> d 0 2 In A . (3.24) sin 0/2 sin 0/2 c 0 0 . min

Assuming only the presence of electrons and the alpha one obtains for a from (3.11) 33

->■ cl n a = f dv ' f (vf) -77: u Av

ZH ^ £e(v') + _> In A --- ^-- u d v 1 (3.25) m mIT e H

“)■ where u = v - vf and f (v*) is the distribution e

function for electrons. We shall use the Maxwe11-Bo 11zmann

distribution

.. Y 3/2 -v v' f (v) N {—) e T (3.26) e e 7T where Ng is the electron density and Y 2KT

Consider the integral

f (v’> e____ dv' (3.27) 1 v-v 1

This integral is worked out in Appendix I and has the value

6 Erf < J?It v where Erf denotes the error function

Since

-y -y 9 -> V - V — |v - v -y -y 9 v v - V 34

one has the relation

91 f (v’) e______(v - v’) dv' | , | 3 9 v v-v

Thus one obtains for the first order F.P. coefficient

. „ 2 4 91 AwZh e In A m mTI e H c 9v (3.28)

91 - r- 9 v whe re 4ttZh ------In A (3.29) m m„ ~ c e H

From (3.12) the second order F.P. coefficients are given by

J'f (v') dv1 u(Av.)2 dfi ii e dM l and from (3.15) we have

2 m (Av.) 4 u 2 sin 2 0 / 2 sin 0/2 6. . + l (“) H i , 1

4- cos^(p cos20/ 2 6. „+sin2(J) cos20/ 26. i,2 T i,3 so that

, _ 2 4 4ttZ e f (v’) dv’ H cos 0/2 I b ii sin 0/2 35

Here again one must introduce the cut-off and one gets

m ^ E b —— r i ii mH 1

ui . — F N - Erf (3.30) m„ e v 2KT

Now since I Ij(v) one can write

- 3Ii £ a . v . - r v . —i i l 3 v

-Tv 3 v 1

r2m 2 ! e -yv N F - Erf e v 2KT ttKT

(3.31) which shows that because of m « one has e H

^ E b . . < < E a . v . (3.32) i i i 11

We then obtain for the rate of energy loss

dE H ~ v ' mH Z ai Vi IT 1

dEH 4"ZHe4 , . „ 1 me V ■, = ------In A N — Erf -yv d t m c e v 2 KT - 2 2 KT

(3.33) 36

The range R will be then given by (2.20) which on simplification gives the following result:

(KT)2 1 x dx R (3.34) m 2 4 N f (x) e itZt1 e In A e H c

where f(x) —— xe X - Erf(x) /tt

_0 and / KT

3/2 (KT) Here A 2 e Z 7T N H /

Numerical results for heavy particles with MeV initial energy and different plasma temperature are discussed later. However the above formula should be corrected theoretically at least as discussed in the next section

3.4____Scattering with soft photon emission:

In the usual consideration for Coulomb scattering when one uses (3.21) as the expression for scattering cross-section one uses more or less elastic type of description of the process i.e. the electron gets deflected and nothing else happens. However this is not entirely true. Accompanied with the change of motion of the electron as can be expected classically there is always some photon emissions of however low frequency they may be. 3 7

Fig . 4

Feynman diagram showing photon emission during scattering. Cross denotes the external field of the heavy charged particle. 38

i o \ This point has been emphasised by Feynman very explicitly. 19) In fact the famous result of Bloch and Nordsick is that in quantum electrodynamics the elastic (i.e. radiationless) scattering amplitude is zero. The scattering cross-section incorporating this aspect has been deduced for the first 20) time by Schwinger in his work on quantum electrodynamics.

Although this is a relativistic derivation it has a nonzero value in the nonrelativistic limit. The cross-section can be also deduced more simply from Feynman rules - see

Appendix III for this - as has been done by Bjorken and 2 1) Drell . The Feynman diagrams to be considered for this purpose are shown in fig. 4.

And we have for this process

da ( da ^ max 4 u .20 In (3.35) d^ dft 3 ~Z Sln 2 min c ph

da Here is the fine structure constant 137 5 dV,

is the Rutherford cross-section as given by (3.21), c the velocity of light and to and 0) . denote the max mm maximum and minimum frequencies of the photon emission. As one takes the limit 0) . 0 one sees that the scattering mm cross-section diverges to infinity which is the "infra-red" catastrophe of quantum electrodynamics. The divergence associated with 03 00 is the "ultra-violet" catastrophe. max 39

To get over these difficulties one again has to introduce

some cut-offs which we discuss below. For the present we

shall write

03 max K (3.36) c 03 . mm

Because of appearance of c in the denominator in (3.35) one may wonder that this is relativistic so that one should set it equal to zero in the nonre1ativistic limit. However % this is not correct as calculation shows that one gets m^c^ in the final expression which is 0.5 MeV quite within the range of the energy we consider. The fact that numerical contribution of this term is not significant - unless one uses very large cut-offs, i.e., large values of

InK-is due to the smallness of a . We now calculate the c F.P. coefficients according to (3.35). We note that 2 q because of sin — there is no need to introduce small angle cut-off in this case, i.e., the Coulomb logarithm does not appear now. However there is another logarithmic term InK due to the quantum electrodynamic cut-off c necessary. One has here

lfi zt e4 r dC7 » ^ 16 a H 1 T, u / -77c u Av d£7 —— ------7 In K — (3.37) diZ 3 2 c u m m„ c e H

72 4 f (v ' ) (v - v ' ) 16a ZH e so that tt In K —------dv* . 2 c m m _ _ c I v - "*V » I e H

v - v -y -> Since v - v V - V 9 v 40

one has 16a Z.2. eA 91 H In K (3.38) P m m _ _ c c 9 v e H with If (v ’ ) v - v’ dv ' e

The integral 1^ has been calculated in Appendix II and has the value

N (h3/2 I Z2 = e 7T with ^ (v) 3/2 v Y

iKv) = —— e + /tt Erf(/y v) (v2 + y— ) /y '

This will give

72 4 -> -> 16a ZH e - 3i2 a . v 3 2 m in _ _ c ln Kc V ‘ 7^9 v e H

16a Z2 e4 H. 2 Ne(i>3/2 ln Kc(-I+ lp' (v)) m mTT c e H (3.39)

In this case we obtain for the second F.P. coefficients the following result

72 4 16a H e ,Ys3/2 , „ , H £ b (3.AO) ii — 2 2 Ne(P ln KcJ mH c which shows again because of the m << mIT 6 e H

-> \h 2 b..| << a . v 41

The rate of energy loss in this case is given by

"TT - m E a . v . d t H . x 1 1

72 4 16a ZH 6 -£ In K ^ ' m c /? C V e „ 2 4 32a ZH e 9 y KT — In K . 2 2 v c 3 /tt m c e

;--- m v - 2 e /fa v / / m v meV ~ 2KT / e 2KT 6 "n * 2KT Erf 2KT J J

( 3.41)

To incorporate this process into the total rate of change in energy one may use the total scattering cross-section as the sum

da f da + d £2 c l >

This would imply that one simply sums the F.P. coefficients obtained for the respective cross-sections or equivalently the rate of energy loss is given by the sum of (3.33) and

(3.41). One thereby obtains the following expression for the range

m c e m N In A _2 4 e e c Z e 2m c ^ e -X Erf(/X)' L)flf^x^7r+(1-i2Erf(^)'

( 3.42) 42

wh ere 20 and KT

One sees immediately that the contribution of the second expression in the denominator which corresponds to the photon 1 emission will be very small due to value of the 137 fine structure constant unless one uses large values for

InK, i.e. for the cut-off:

max (3.43) min

However the logarithm function is very slowly varying so that InK does not depend so much on the specific choiceof co and w . . In fact this problem max mm of cut-off and its choice is a major defect of the present state of quantum electrodynamics and the quantum field theories. One possible choice could be given by the instantaneous energy of the heavy particle and plasma binding energy E . Regarding this latter we B 22) note that Bagge first utilised the expression

h (3.44) 4ttN e e

Here is the Debye length of the plasma. Bagge used this for a possible modification of the Bethe-Bloch formula for electron penetration in plasma in order to obtain better consistency with the experimental results.

We shall discuss this type of theory in detail in the next section. For the above choice we note that one obtains 43

KT 3/2 (3.45) 2 /4tt N

to be used in (3.42). However numerical calculation shows that this gives insignificant change for range as compared with (3.34). It is interesting to note in this respect that in connection with a renormalisable field theoretic 2 3 model Pauli ) has conjectured a finite cut-off momentum P for quantum e1ectrodynamic calculations which is of the order

1 In a 137 m c e

This would imply for a frequency cut-o f f

1 In K ~ c 2a

24) This argument has been independently supported by Landau where analyticity of the photon propagator is involved. 25 ) This has also been discussed by Low . Only this type of higher cut-off gives some difference with the previous

F . P . f o rma1ism.

There is another way of getting over the infra-red divergence, i.e., adding the higher order radiative corrections and cancelling the infinities. In that case the scattering cross-section will involve other terms.

The ultraviolet divergence will however remain. We mention that this way of handling the infinities, the so 44

called "renormalisation" technique, is not universally accepted by physicists and one may say that at the present stage of quantum electrodynamics one has just reached a sort of "peaceful coexistence" with these unsolved difficulties.

3.5.____The "collective" model for energy loss :

We now discuss a model which is very similar to the calculation of energy loss and range as performed in nuclear physics. Consider the motion of the heavy particle which we take as the x-axis and a point P situated at a distance b from its path. Then b is the impact parameter. An electron situated at P will experience a resultant average force in the direction of normal to the path (fig.5) which we will denote by . The component of the

Coulomb force will then be given by for the position shown

F V cos 0 1 2 r

ZHe2b 3

If this force acts for a time At the momentum gained by the electron Ap will be

Z e 2b _H___ At Api 3

zHe 2 b AX 3 v r where AX is the distance travelled by the heavy particle.

A X One has —— = v the velocity of the heavy particle. Thus 45

Fig. 5 b impact parameter

Fig. 6 46

the total momentum gained (Ap^)^ by the electron at the position P is

+ 00 vZ He 2ub dX (AP± ) 3

2ZHe2 b v

This will correspond to an energy transfer AE to the electron

972 4 2ZHe AE (3.46) m b 2 v2 e

Now the energy loss -AE per path AX of the heavy particle will be the sum of the energies gained by all the electrons situated in the cylindrical volume as shown in fig. 6 which has then to be integrated

b max 4uZ2eA H N b db AX Aeh e b,2 v 2 m e bmin giving

ZHg4 max 4ttN In (3.47) e 2 m v min e

Here b and correspond to the maximum and max min minimum values of the impact parameter. For one can use the classical value

V min m v e which is obtained by simply noting that the minimum value of the impact parameter corresponds to the maximum energy transfer (3.46). Classically the maximum energy 2 transferable to the electron is 2m v e

For b one has in plasma physics a natural max cut-off which is given by the Debye screening radius :

b A max D

(3.48) / KT 47TNee

This gives then from (3.47) for the energy loss per unit path

/ r, 2 4 4TrZTle / m v H f Ge /KT \ -----2- Ne ln l - 3 m v V Z„e3 e H P™e J

2 4 mTT 7TZ„e // m — —f- N m I -s. me EH 6 ^ mH NM e 6Zv 2 H e H (3.49 )

The range will then be given by the following formula

2 m r\ R = -- Ei (In (AE )) (3.50) 2KT m o e whe re

x ■h X ttN ZI e6 e H and Ei is the integral logarithmic function defined as

X f t E i ( X ) dt 48

One should note that although the logarithm function In

varies slowly with respect to its argument, Ei varies

very sharply for a small change in its argument.

In the above derivation one might wonder about the

effect of the high thermal velocity of the electron on

energy absorption. But this thermal velocity is random in

character in plasma and drops out as may be easily seen on

averaging. For let the electron have a momentum p^

due to the plasma temperature - one notes that p^ is

random in character. The effect of the interaction adds

to it a momentum p^ so that the energy gained AEg

by a single electron is

-> -> 2 (pr + px> 2 m

that the average value of the energy AE is given by

AE = (AE ) a v s av

F± l -> -> 2m~-- + m-- (P*r • *±P . )av e e

which is utilised in deriving (3.46). 49

3.6 Relation to the Bethe-Bloch formalism and Bagge's

modification

We shall now discuss the relation of the above model

with the Bethe-Bloch formalism which is normally applied for

penetration of charged particles in solid matter. Histo

rically the theory of energy loss of heavy charged particles

in passage through ordinary matter was first studied by 2 6) Bohr taking the classical picture. He calculated on

the basis of the model that the electrons in matter, to which most of the energy is transferred by collisions, are

elastically bound to their equilibrium positions. Assuming

one electron per atom his result is the folio wing

/ k m v dEH 47TZ^e4 / e In - *5 In ( 1- m v ' 2ttvZ e2 2c e H

(3 . 5 1) where is the velocity of light, is the cons tan t

2e~Yc

1.123

Yc being the Eu1er-Mascheroni constant y = 0.577, and V the characteristic frequency of vibration of electrons in atom. In the nonrelativistic limit this will simp 1if y to

k m v' dEH 4TTZ^e4 _____ e N In A (3.52) dX 2 e m v 2ttvZ e2 7 e H 50

The problem of energy loss according to quantum mechanics 2 7 ) was first studied by Gaunt . He replaced Bohr's considerations of the atom as a classical oscillator by the quantum mechanical picture. His result is almost the same as Bohr's where the characteristic frequency V has to be replaced by ionisation potential, i.e. ,

hV Ionisation Energy of the Atom I where h is the Planck's constant. In his calculations he considers only the interactions with atoms whose distance b from the path of the incident particle, i.e., the impact parameter, is sufficiently large. He has conjectured that the divergence associated with making b infinite would cancel out with that of making b zero and he sets for the minimum value of b the classical expression

min m v^ e

Gaunt has not given any reason for his classical type of calculations for small values of b which may differ considerably from the quantum mechanical results.

2 8) Bethe was first to calculate in a consistent manner using quantum mechanics under the approximation of high velocity of the incident particle the energy loss formula which is

2 4 2 4ttZ H e (2) m p v ---—— N E f In ----- 5 (3.53) m v n n h (V - V n) e n 0 51

Here the index n denotes the different quantum states

of the atom, the sum is over all quantum states, the

number of atoms per unit volume, f the quantum mecha

nical oscillator strength, \)^ the frequency of the nth

state and that of the ground state. Furthermore

the symbol (2) means that there is a factor 2 in the

argument of the logarithm if the deflection of the incident

particle due to its interaction with the atom can be

neglected, i.e., for the case of heavy particles like the

alphas, protons, etc., whereas for the case of electrons

there should be complex expression in place of (2) which

takes into account the straggling effect. For the

approximation of classical harmonic oscillator the above

formula reduces to

4/ tt Z<7 2 e 4 2m v2 H e N In (3.54) e hv

As is easily seen this formula differs from Bohr's formula , „ 2 4ttZh6 (3.52) by the factor —j-j— in the argument of the

logarithm which points to some difference in principle due

to the appearance of v .

The next step in the development of this theory was 29) 30) taken by Bloch * who analysed the assumptions under

lying the calculations of Bohr as well as of Bethe specially with regard to the use of Born approximation. He derived a result from which both of these previous calculations 52

could be obtained under special approximation. His

assumptions involve firstly that the momentum change of the

incident particle is always small compared to its initial

momentum and secondly the velocity of the electron in atom

is smaller than the velocity of the incident particle. For

the nonrelativistic case Bloch's formula is

2 4 2 2 dE 4ttZ e r 2mv Z e - dX" = -----2 NA z fn tln h'(V~-Vn) + -MO-Re^l + i m v n ^ n 0

(3.55) where ^(Z) is the logarithmic derivation of the Gamma

f unc tion

iMZ) = ^ In T(Z)

2 77

and Re denotes the real part of the complex value. One

has in particular

One sees that Bethe's result is obtained for small values

Z e H o f fil In fact for only the first hv hv

term in the bracket of (3.55) survives which is same as

Bethe's formula (3.53). On the other hand for large Z e ^ He values of -^ ■ - one obtains

, ZHS \ ZHe Re i|>( i + l -j— J = In -j— 53

which means that (3.55) goes over to Bohr's formula (3.52).

For practical computation in the nonrelativistic case one simplifies (3.55) by the sum rule

L fQn In h(vn v0) za ln eb n where Z. is the atomic number of the matter and A B average binding energy. Bloch has shown that one can write according to Thomas-Fermi calculations

E_ = I . Z. (3.56) B A where I is an experimentally determined constant. Bloch has given the value I = 13.6 eV.

Since one can write (3.55) in the simplified e A A form for heavy incident particles

4ttZ^ e4 2m v dE H. e N ln (3.5 7) dX e m v e which is usually referred to as the Bethe-Bloch formula.

In connection with the penetration of high energy 3 1 ) electrons in plasma Bagge and Hora have used the following estimate for the plasma binding energy

2 e E B (3.5 8) 'AttN e KT 54

For low density plasma E can be measured by the decrease B 32 ) in ionisation energy and this shows that E has the B property of "plasma binding energy". This will then give

, 2 4 2 2m v dE 4,ZHe e N In (3.59) dX e 3 m v e e comparing (3.49) one sees that (3.59) differs in the argument ZH of logarithm by a factor — • Although due to the slow change of the value of logarithm function due to change in its argument the numerical results may not differ strongly for low Z in applying (3.49) or (3.59), the difference is of importance in principle.

We again emphasise that for the derivation of the

Bethe-Bloch formula or the Bagge's modification of it, it has been assumed that the electron velocity in plasma is smaller than the velocity of the incident particle. This is true for MeV electrons as incident particle for plasma temperature in keV region. However for MeV heavy particles such as alphas, protons etc., the velocity is of the same order as of the electrons in plasma of temperature keV range .

Since the derivation of our collective model does not depend on this restriction, one should apply this result for the case of plasma.

It is interesting to note in this connection that for the penetration phenomena in the ordinary matter in 5 5

certain cases of low energy region the deviation of results based on classical mechanics from experimental data has led

to the supposition of nonvalidity of classical mechanics

for such conditions. Indeed the quantum mechanical 33) calculations give much better agreement. However Gryzinski has traced the source of the trouble which is the neglect of the effect of orbital motion of atomic electrons. He calculated this again according to classical mechanics by

taking this into account more accurately and has obtained results in good agreement.

3.7 Numerical discussions and conclusions

The numerical result for the ranges in different cases according to F.P. formalism and the collective model are shown in fig.7 - fig.12. One notes that the two models give entirely differentresu1ts not only numerically but also phy sically. According to the former the range first decreases and then increases as the temperature rises whereas for the other case the range slowly drops. In the absence of direct experimental results it is difficult to form an opinion regarding the validity of models. In all works of plasma physics , to the knowledge of the author, one has used always the F.P. type calculations and utilises the following behaviour of the range

3/2 R ~ T (3. 60) 56

This result seems to be approximately valid in the region

1-50 keV plasma temperature of our calculations. The fact that the range increases with the’ plasma temperature has been utilised by Brueckner and Jorna 3) to construct an ignition model of nuclear fuel of highly compressed spherical mass whose dimension is of the order of alpha particle range. They assume that by implosion technique one may achieve shock heating at the centre of the implosion, i.e., centre of the fuel thus initiating thermonuclear ignition there. As the temperature rises locally the hot alphas would travel further out to the adjacent cold fuel which would then cause it to ignite.

The conjecture is thus the propagation of a spherically expanding burning front in the fuel.

O £ \ We also mention that Winterberg gives the 4 range value of 10 cm for electrons with initial energy several MeV striking a hydrogen target at thermonuclear 2 2 3 temperatures with density 5 x 10 particles per cm .

We shall however assume in our further calculations the validity of the collective model for the following 37) reasons. One measurement which is available is the absorption of high energy electrons in - plasma - it has been observed that the penetration depth of 2 MeV electrons in CD^ - plasmas are less than that for the corresponding solid state by magnitude of order one. 31) Bagge and Hora have successfully explained this by the modification of Bethe-Bloch formula as stated earlier. RANGE of 2 .8 9 MeV ALPHA in H3 - PLASMA Ns SOLID STATE ELECTRON DENSITY ■ 2.82 x I023cm' No PLASMA ELECTRON DENSITY Fig.

7 sujo

36 ud ^ 10 57

Temperature in EV RANGE of 14.7 MeV PROTON in DHe - PLASMA Ns SOLID STATE ELECTRON DENSITY> 5.S x 1022 cm Ne PLASMA ELECTRON DENSITY Fig.

8 58 Temperature In RANGE of 3.5 MeV ALPHA IN DT- PLASMA Ns SOLID STATE ELECTRON DENSITY 5.8x10**071 Ne PLASMA ELECTRON DENSITY COLLECTIVE MODEL SLUO Fig.

9 Uj

06UD^j 59 Temperature in EV RANGE IN H3 PLASMA OF SOLID STATE DENSITY ALPHA OF 2.894 MeV. saJD

Fig. ni

10 39NVd to 2 < CL < -j TEMPERATURE IN EV 60 RANGE OF 14.7 MeV PROTON IN stuo

Fig. NJ

11 39 NVd UJ > uj LU q h* K UJ CL < D I- CL < _J : 61 3.5 MeV ALPHA PARTICLE RANGE IN OT-PLASMA OF SOLID STATE DENSITY st-UD

Fig. Ml

30NV&

12 62 63

Now the F.P. calculation gives penetration depths of much higher value than that of the collective model -- for

3.5 MeV alphas in DT plasma of solid state density 22 -3 = 5.8 x 10 cm and plasma temperature 10 keV one

gets according to the former the value 1.33 cm whereas for _ 3 the latter the value 4.3 x 10 cm, i.e., difference is by

a factor of magnitude of 1000. This is also true for

other cases. One would conjecture that the values should be close to the solid state case for such high densities.

This decides in favour of Bethe-Bloch type calculations.

3 8 1 Finally we mention that since it is known

that the usual Boltzmann equation holds for low density

gases only, it is theoretically objectionable to apply the

F.P. formalism in its present form for the high density plasma. For example the Maxwe11-Bo 11zmann distribution

for the electrons may become severely altered for such high densities which would then imply a significant change

in F.P. coefficients. 64

CHAPTER IV

INTRODUCTION:

In this chapter we shall study a model of thermonuclear fusion plasma under inertial confinement.

A detailed experimental view of a possible fusion reactor based on such a concept has been described by Nuckolls 39 et al ) . We mention that for the case of magnetically confined plasma, e.g., in the tokamak, the particle densities are of the order 10^ - 10^ which is relatively easy to achieve. However, for sufficient energy produc tion one has to confine this for a relatively long period which has posed great difficulty because of plasma instabilities etc. On the other hand the prospect of laser-driven fusion seems to be promising at present. 6 5

4.1_____Model of a high density expanding plasma

The fundamental concept for the inertially confined plasma is that of the production of high density very hot plasma in vacuum which may expand adiabatica1ly under its own inertia. This is achieved by irradiating a solid target of very small dimension by pulsed giant laser radially from all directions. This idea goes back to Basov and after a dormant period has been again under intensive attention by various groups of plasma establish- 40) ments. Dawson has presented some theoretical calcula tions for such a plasma. This self similarity model has also been discussed by Hora 4 1) starting from hydrodynamic equations. Since the rate at which thermonuclear reactions take place is given by

R = n^ n^ where (n^, are the densities of the reacting nuclei and

the Maxwell averaged cross-section, one sees immediately that one must have very high ion densities to obtain suffi cient energy release. Now depends on the plasma temperature T and as can be seen from the data as T drops

falls sharply. This aspect is very important since the adiabatic expansion of the plasma could cause enough cooling.

It seems that in all previous published works on fusion yield calculation this point has been neglected excepting 42) 43) those of Hora and Hora and Pfirsch • Numerical calculations show that for break-even energy to be achieved, i.e., where fusion energy released just equals the input 66

laser energy one requires for the ion density to be at least of the order of the solid state density or even higher.

Now the experimental process of the production and measurement of this high density plasma is very complex. 39 ) Nuckolls et al have discussed this in detail. One uses the implosion technique to reach density of the solid state and higher. First the surface of the microtarget is vaporised by the incidence of a few pulsed laser radiation.

This transformation is very fast and is of the order of nanoseconds. One obtains a plasma with solid core. Further laser irradiation would result in a very hot plasma atmos phere where the enormous rise in pressure would cause a compression of the core. Then the heat by thermal conduction would transform the core in a very hot dense plasma where nuclear ignition would take place. Subsequently a thermo nuclear reaction front propagates outward. The mechanism of laser light absorption by the plasma is very complex and we shall not deal with this aspect. We mention that in order that light may penetrate the plasma the plasma frequency should be less than the light frequency. In our model we assume that the transformation of the target to the plasma state is instantaneous and the laser energy is deposited uniformly throughout to achieve a high temperature. 4.2 Mathematical Formulation of the

Thermokinetic Expansion Theory

Consider a spherical mass of plasma of radius R with total number of particles N. We shall assume the density to be uniform within this sphere and imagine that the whole thing expands uniformly. Now during expansion the internal energy given by the temperature T, which microscopically is the kinetic energy of the random motion of the electrons and the ions, is converted into the macroscopic kinetic energy of expansion. This macroscopic energy of radial expansion is given by the total mass M of the particles multiplied by some mean velocity squared. One notes that this is equivalent to treating the plasma as if it expanded with the velocity of the outer shell and had a certain effective mass M because all the parts do not expand at the same rate, the part which is near the centre expands slower than the outside.

Now the rate of change of volume V is

dV 2 dR 4 7T R (4.1) dt dt

The work done by pressure is where P denotes the pressure. If one equates the rate of increase of kinetic energy of radial expansion one obtains 68

If we assume the density of the plasma to be uniform and also the velocity to increase linearly from the centre to the 3 edge, one has then M = yM. One has therefore*

2 dR. 1 3M d ,dR, 2 P . 4ttr (4.3) d t 2 5 dt ldtj

If N. and N denote the total number of ions and l e

electrons (N = N. + N ) one has furthermore * l e

KT P 3 (N . + N ) (4.4) l e 4ttR2

If W be the rate at which energy is supplied by laser one

has the equation of conservation of energy

y(N + N.) =-47TPR2 ^7 + W (4.5) 2 e l dt dt

Here the left hand side represents the thermal energy of

the plasma. We assume that the time for which the laser

works is some picoseconds during which the solid target is

plasmatised and the temperature is raised to with

radius of the spherical mass R . The expansion of the

plasma takes place from this stage so that one can set W = 0.

Then from (4.4) and (4.5) one gets

KT dR 1 _d_ (KT) (4.6) 2 dt R dt

which gives on integration * S e e note on p.124 69

T = T o (4.7)

combining this with (4.3) and (4.4) one can write

KT dR 3M d (dR 2 3(N.+N ) l e R d t 2 . 5 dt ldtj

which on integration and utilising (4.7) gives

o M o [l-(-^)2]

where the dot signifies the derivative with respect to time.

Now for a plasma with average ionic charge Z one may use

N - Z N . so that e i

M = m.N. + m N - m.N. li e e li

where m. is the average ionic mass (m^>>mg) and this gives

5KT (l+Z) R (4.8) R2 = Ro2 + ---m5------. [l-(^)R 2]

In our following calculation we shall use Rq = 0. 70

4.3_____Fusion Yield in Adiabatica11y Expanding Plasma

Consider now an adiabatically expanding spherical plasma which has radius R(t) and respective reacting ionic densities n^(t), n^ (t) at instant t of expansion. All

the entities at t = 0 will be denoted by subscript o.

Since the rate Rp of fusion reaction per unit volume is given by

RF = n p(t) n 2(t)

one has for the total energy E released by reactions K

00 E = £ / dt / n(t) n„(t) <0v> dx dy dz (4.9) K r ^ , \ 1 Z t=o R(t)

where £ is the energy release per reaction and the

integral over R(t) means over the entire spherical volume of R(t) . We assume that the respective ions have 50-50

ratio so that if n^=n^(t) denotes the total ion density nl=n2=^ni(t)' 0ne can therefore write

n? (t)

nl(t) n2(t) = ~k---

where A = 4 if ions are of different species, e.g., HB

reaction and A = 2 if ions are of same species, e.g.

DD reaction - in the last mentioned case the same ion cannot

react with itself. If E denotes the total laser energy to o

produce the plasma at initial temperature Tq the fu sion 71

efficiency G is given by

2 F 4 7T i n ' RJ —- dt (4.10) A

where the uniformity is assumed through the volume of the

plasma. One can transform this integral to a computable d R form by using dt = -g- where R is given by (4.8) . Further more, assuming not very much depletion of the ions due to 3 o 3 reaction one can write n.R - n. R so that one gets 1 1 o

( o,3 G (4.11) ( R ; A 5KT (1 + 4) o

Now the dependence of on R is expressed through the

adiabatic expansion law (4.7)

2 2 TR = T R o o

so that

(T) = (T(R)) (4.12)

The integral in (4.11) has an apparent singularity at

R = Rq. This can be removed for numerical calculation by

the substitution 72

R = cos0 (4.13) K

This implies further

T = T cos20 (4.14) o /5KT (1 + Z) R = -- -jJL----- s in0 (4.15) v^T

n.(t) = n? cos^O (4.16) 1 l

where for the last expression we assume n . R n °. Rp 3 l 1 o For fusion efficiency G we obtain

7T / 2

G / mt R o c o s 0 d 0 5KTq(1+Z) O (4.17)

The dependence of on the parameter 0 is through

the temperature

(t) (T cos 0) o

Since cannot be expressed as a simple function of 0

one can only compute (4.17) numerically. The numerical

results are discussed below. We simply note that for

computation an increment A0 in 0 would imply a change

AT in T given by 3.

AT = T [co s 2(0+A0)- co s 20] a oL J

= -2T sin0 cos0 A0 (4.18) o

= -2T t an 0 A0 which is the adiabatic law for temperature decrease. 73

4.4 Contribution from alpha particle reheat

One important aspect in the above calculation is that of the absorption of alphas or other charged particles produced in the reaction by the expanding plasma itself. The averaged value depends sharply on the temperature. Now the absorption of the hot alphas would the raise the temperature givingalarger contribution to (4.18). This is known as alpha particle reheat and one would expect better reaction gains due to this. This is a highly complex nonlinear process and one can take this into account num erically in an approximate way only. The numerical results are explained below — we simply note here that we assign a probability for absorption in the following way. Suppose when the plasma radius has a value R and plasma temperature T, the range of the alpha is R ^ . Then the probability P^ of

its absorption is given approximately as

R P (4.19) a R+R a

Now the rate at which alphas are generated in the plasma

2 Rateof n ^ alpha production = —- .V x No. of alphas per reaction

The rate at which the corresponding reaction energy carried by the alphas will be absorbed by the plasma is then given by 74

2 n . . . V (4.20) a F ,a A

where ^ is that part of the fusion energy released

which is carried by the alphas. Now the effect of this would

be to alter the simple adiabatic law (4.14). However in

this case one cannot integrate and find an analytic

expression of T as a function of 0. This makes the

calculation complex.

We start ab initio in this case to show the necessary

changes involved in the previous calculations. To take the

alpha particle contribution to temperature change accurately

one should take into account the ionic depletion. The

reaction gain G is as before the ratio of total energy

released to the input laser energy so that

x Total Number of Reactions

dt Iff —— < G v > dx dy dz R(t)

Here n^ is the average reacting ion density at the

particular instant considered when plasma radius is R(t).

Assuming again uniformity throughout the volume one can write 75

^ n . F 4 TT d t R < a v > A t = o

n < a v > (4.21) (f>o -nr“ dt t = o

Using substitution (4.13) as before one obtains

TT / 2 2 . ^ ' ni<°v> ae (4.22) h v° o A co s ^0 ®

Here 6 is a complicated function of t. To evaluate we express all the entitites under the integral as functions of Q. Then for an increment A0 one has to calculate such functions changing according to

dn . n.(0 + A0) = n.(0) + Q A0

T (0+A 0) = T (0) + (Jq-)0 A 0 (4.23)

0 ( 0+A 0) = 0(0) + (^|)0 A 0

The dependence of on 0 is through the temperature

T .

Let us now construct the fundamental equations for thermokinetic expansion. Equating the work done by pressure to the increase in macroscopic kinetic energy one has 76

(4.24)

where the pressure P will be given by

3(N.+N )KT 1 e (4.25)

Here N and N denote the total number of ions and i e electrons present. Since the electrons do not change by the reaction one has always N = Z*N? where N? is the e i l total number of initial ions present. Now the alphas generated by the reactions will be mostly absorbed in a high density plasma so that although the species of ions change their total number will be approximately constant. We may thus write

N. + N - (1 + Z) N° (4.26) l e i

Then (4.24) and (4.25) give on combination

(4.27)

where we have again used M = — M - m ^ N ° . In terms of 0 equation (4.27) becomes

5(1+Z)KT l m . R o o (4.28) 77

This gives the initial value 0 for 0 = 0 as o

5(l+Z)KTo (4.29) o m . R2 1 o

Using this one can write

02cot0cos20 . „ 9 d 0 —— — ------0 cot0 [sec 0 + tan 0 ] (4.3 0 ) d0 ro 0

where at 0 = 0

(—) 0 ld0J

The second equation for thermokinetic expansion is

that the rate of change of the total internal energy should be equated to the rate of change of macroscopic kinetic

energy and the rate at which the alphas are absorbed. Using

(4.20) one can then write

d d (|(N +Ne)KT) (} M (—) 2 ) dt d t

2 . ^ n . R l + (4.31) F, a R+R A a

Here n . denotes the density of the reacting ion species. l

This gives utilising (4.24) and (4.26) 78

3 ,, tt dT 3(1+Z)KT dR 2 (1+z) K dT ” R dT

ni (_R_} 3 + £ (4.32) F,g R+R n?(1+Z)A Ro

In terms of 0 this is

n . < o v > " F , a 1 _l______-2KT t an0 + (4.33) 1+Z l+^-cos 0 n°Acos^ 0 l o

One needs another equation for n.. Let N. denote the i l , r total number of reacting ions in the plasma at time t.

Since they change only due to reaction one can write

2 dN . i , r n i 4 3 ~Y .^-TTR (4.34) d t

Note the factor 2 in the denominator instead of A in this case. It is the same whether the same species of ions are reacting (e.g. DD) or different species (e.g. HB^). Now

N . l r

Therefore

dn . __l ~ni dR dt R dt

3n. t an0 0 l

The first term here denotes the change due to reaction and 79

the second the change in density due to radial expansion.

In terms of 0 one can then write

dn. 1 n * IF = “ 2 “1------3ni tan0 (4>35)

Thus the set of relations (4.23) can be evaluated numerically by using (4.30), (4.33) and (4.35). One can then compute the efficiency. G by the integral (4.22).

We finally note that for extremely high density plasma, say 10^ times solid state density or higher, one has to take into account the ionic depletion more exactly, i.e. the relation (4.26) may not be a good approximation. 80

4.5 Numerical Results and discussions

The reaction gain G evaluated numerically for DT 11 as well as HB thermonuclear plasma are shown in fig. 13

- fig.23. for different values of initial plasma volume

Vq , input laser energy and initial ion densities n^.

Let us first consider the results without reheat. The curves for a fixed ion density and different volumes have a common tangent line. This line determines the optimum gain

G t for a given input energy. It has the meaning that given an initial energy one can always choose a certain initial plasma volume for a fixed initial ion density so that one obtains the highest possible reaction gain. This gain is given by the ordinate on the curve for the abscissa

Eq. Numerical results show that the following law holds for both the cases considered

(4.36)

where E is the breakeven energy — which is defined as D that input energy for which ^Gpt = ^ — for t^ie s ° 1 i d

g state initial ion density n^. One should note that this does not correspond to the maximum gain G for a r max certain volume and ion density. For let us consider the numerical values for HB^ plasma: 81

23 Initial Ion Density = 9.4 x 10 cm

E° (eV) G V (cm3) Eo(J) av o

6 x 106 99.7 x 103 1.1039 x 10-2

7 x 106 116 x 103 1.7332 x 10"2

1 x 107 166 x 103 3.0718 x 10"2 1 x 10~4

1.5 x 107 249 x 103 3.3078 x 10"2

2 x 107 332 x 103 2.6302 x 10"2

6 x 107 99.7 x 103 2.3784 x 10”2

7 x 107 116 x 103 3.7340 x 10-2

1 x 108 166 x 103 6.6179 x 10-2 1 x 10"3

1.5 x 108 249 x 103 7.1264 x 10"2

2 x 108 332 x 103 5.6665 x 10"2

6 x 108 99.7 x 103 5.1240 x 10"2

7 x 108 116 x 103 8.0447 x 10~2

1 x 109 166 x 103 1.4258 x 10-2 1 x 10~2 9 1.5 x 10 249 x 103 1.5353 x 10"1 9 2 x 10 332 x 103 1.2208 x 10_1 82

Here E denotes the average energy per particle initially.

It is related to the initial temperature Tq and input energy EQ as:

I KV (1+Z)n:v i 0

Now for the same initial temperature Tq but different initial volume Vq one finds that the following relation holds numerically:

log G = A log Eq 4- B

where one has A = — .

The constant B depends on Tq and has the following values :

E° (keV) B av

99 . 7 -4.21645

116 -4 .04285

16 6 -3.84594

249 -3.87249

332 -4.01369 USION OF DT WITH INITIAL ION DENSITY 5 .8 x l022cnf U109 Fig.

uoiiooay 13

Input L a s e r E n e rg y EQ ( jo u le s )V ,Q In itia lV olum e(c m3) Upper Dashed Line Gives Gains incorporating Reheat 83 Lower Curves Without Reheat and Depletion FUSION OF DT WITH INITIAL ION DENSITY 5.8x10 cm Fig. UIDQ

UOJIOO0U 84 D Q b sz u L o D. CO o c c o o L_ CL o V- o c CO $ O 13 I FUSION OF DT WITH INITIAL ION DENSITY 5.8 x 10 o G Fig. UIDQ

UOIIDOa^J

Input L o s e r E n e rg y EQ (jo u le s ),VQ In itia l Volum e (cm3)

U pper D ashed Line Gives Gain Incorporating Reheat 85 Lower Curves without Reheat and Depletion FUSION OF DT WITH INITIAL ION DENSITY 5.8x10 cm x

/ O

/ Fig. UjDQ

UOljODa^J

Input Laser Energy EQ (jo u le s), VQ Initial Plasm a Volume (cm3) Upper Dashed Line Gives Gain Incorporating Rehect 86 Lower Curves Without Reheat And Depletion FUSION OF DT WITH INITIAL ION DENSITY 5 .8 x 1 0cm Fig.

17 U|

£)

UOll-ODaH CJ

Input L aser E nergy E0 (jo u le s ),VQ Initial Volume (cm3) Upper D ashed Line Gives Gains Incorporating R eheat Lower C urves Without R e h e a Ionic t, Depletion Taken Into Account 87 FUSION OF HB1' FOR INITIAL ION DENSITY 9.4x10 cm

(SOLID STATE DENSITY) Fig UjD .18

UOjiODSfcJ

Input Laser Energy E0 (jo u le, s )VQ In itia l Volume (cm 88 FUSION OF HB" WITH INITIAL ION DENSITY 9 .4 x 1 0 cm Fig. U|D9 CVJ O i x

U0UDD9^j

Input L a s e r E nergy EQ (jo u le s ),VQ In itia l Volum e (cm3)

Upper Dashed Line Gives Gain Incorporating Reheat Lower Curves Without Reheat And Depletion 89 FUSION OF HB“ WITH INITIAL ION DENSITY 9.4x10 cm Fig. u 20

uojiooa^j

Input L aser Energy E0 (joules) , V0 Initial Volume (cm3) Upper Dashed Line Gives Gains Incorporating R eheat 90 Lower Curves Without Reheat And Depletion FUSION OF HB" WITH INITIAL ION DENSITY 9 .4 x 10 cm Fig. 21 UID0

UOj.J.ODa^j

Input Laser E n e rg yE0 (jo u le s ),V0 In itia l V olum e(c rrr)

U pper D ashed Line Gives Gains incorporating R eheat 91

Lower Curves W ithout Reheat And Depletion FUSION OF H 8“ WITH INITIAL ION DENSITY 9 .4 x 1 0 Fig. 22 UJD

UO!|OD0£J

L a se r E n e rg yE0 (jo u le s ),VQ In itio !Volum e (cm3) 92 U pper Dashed Line Gives Goins Incorporating R eheat

Low er C u rve s W ithout R eheat __ Ionic Depletion Taken Into A ccount Here 93

ro E o

N CJ ) o 3

X (cm

<3- O) > I— Volume CD 2! IjlI Q 2 Initial

O Q V

_J t cC CO

ll] z Q (Joules)

I 0 h- E I— < Reheat

5 i—

CD 9

X _j Energy

o LL 00 Incorporating Laser

Gains Input

1 (3 1 1 CJ o O o o ujO0 uojioDa^j

Fig.23 94

The tangent line defining the optimum gain is given by the maximum value of B which occurs for = 166 keV a v

approximately. This defines the optimum temperature Tq^^.

We emphasise that T ° ^t is independent of the initial ion

density for the case without reheat. Numerically

| 10 keV for DT opt KT \ (4.37) 0 110 keV for HB"^

The implication for this is that for a given Eq and n^

one has to choose a particular in order that one obtains

the best fusion gain according to

E0 = | KToPt (1+z> ni vo (4.38)

For such an optimised volume the reaction gain will satisfy

the relation (4.36). Now combining (4.36) and (4.38) one

gets

(2TTKT°pt) 1/3(1+Z) 1/3 G (4.39) opt .1/3, s,2/3 EB (ni)

From the numerical computation one obtains

1.5 x 106 Joules for DT

2.5 x 10^ Joules for HB^ 95

This gives

1.66 x 10 22 n? r for DT l 0 G opt 1.26 x 10'24 n° for HB1 1 l 0

which shows that for successful efficiency one has to reach density higher than the solid state.

We note here that the experimental data for HB is not very accurate — the upper line shown in Fig. 2 gives more optimistic gains. This is about 2 times higher whereas there is no change for Tq^*". In this case the break-even 1 2 energy Eg is 2.5 x 10 J.

We also mention that the above relation for G opt in t e rms of n and R corresponds to PR criterion set 44) up by Kidder. The latter replaces the nr criterion o f 45) Laws on .

One should also note that there is always a limiting gain G^ for t^ie reactions — this is the gain when all the ions are burned up. This value can be obtained from the fact that only ^n^V^ number of reactions are possible for a 50-50 ion mixture so that 96

CFr x hni. 1 im KT°pt n° ( 1 + Z) V,

3KT°pt (1+Z)

|300 for DT

7 for HB1 1

Then one has the restriction

G . <= opt 1 im

However,this is a conservative estimate since the incorporation of reheat essentially changes the concept of

T°pt. For the reheat calculation it is difficult to derive a simplified law like (4.36). Furthermore there does not exist any definite initial temperature independent of the initial ion density where the maximum gain occurs.

For solid state density of HB11 the gains incorporating reheat and ionic depletion are the same as those calculated without the reheat and depletion effect. At higher densities the reheat effect is more pronounced for HB

than for DT. Some interesting numerical values are the

following: DT plasma:

Laser input Initial Initial Initial G energy temperature volume ion density 0, -3v VJ) KTQ(keV) Vq (cm8) n^(cm )

6 -4 , 25 2 x 10° 0.7 1 x 10 5.8 x 10 2.7 x 10 1

3 x 106 1 1.5 x 103

5 x 106 1.7 9.3 x 102

8 -2 „ , ?5 1.5 x 10 .5 1 x 10 5.8 x 10 3.3 x 10

2 x 108 .7 3.8 x 103

3 x 108 1 2.3 x 103

1 1 HB plasma:

Laser input Initial Initial Initial G energy temperature volume ion density 0, -3v E0

27 1 x 107 10 1 x 10~7 9.4 x 10 2.0 X io_1

1.5 x 107 15 3.6 X 101

2 x 107 22 2.8 X io1

27 6 x 108 6 1 x 10~5 9.4 x 10 3.4 X io_1 oo r o — X

1 7 1.4 X 10° 9 1 x 10 10 6.3 X io1

1.5 x 109 15 4.2 X io1 ro DT PLASMA WITH INITIAL ION DENSITY 5.8x10 cm ro (I03 x SOLID STATE DENSITY) INITIAL VOLUME IO'3c E Fig .24 98 99

The very promising point here is the occurrence of the maximum gain at initial temperatures of 1 keV and 1 1 7-10 keV for DT and HB respectively. This would imply limiting gains

3000 for DT

1 im 1 1 100 for HB

Such a temperature implies some sort of ignition process for the fuel. This may seem at first to be surprising because of the low reaction cross section at those temperatures.

But detailed numerical considerations have shown that at such high densities since most alphas are absorbed the reheat contribution is much higher than the temperature drop due to adiabatic expansion so that the temperature gets raised to a point where the reaction cross-section is significant.

One sees from above that for successful energy production one has to achieve much higher ion densities than in the solid state. And this has been problematic experi mentally with the implosion technique. It has been reported that densities of the order of 100 times solid state have been reached. In their scheme Nuckolls et al proposes 4 attainment of 10 by ablation mechanism. Theoretically however there exists another good possibility for this if one makes use of the properties of "non linear forces" as 46) developed by Hora. For the laser plasma interaction one can distinguish two essentially different categories of forces namely the thermokinetic type whose origin is the gas dynamic 100

pressure and the electromagnetic type which is expressed

through (E, H) the electromagnetic field within the media.

These forces of electromagnetic nature depend on the

dielectric characteristic of the plasma which in turn depends

on the electron density. We refer to the relevant literature

for the details but just briefly mention the mechanism

involved. For the case of a perpendicular incidence of a

plane wave on plasma with propagation in the x-axis

direction the time average non linear force density

x-axis can be written as

? ? Ez go 3% f = p_n NL 1 6tt 2^2 3x oj n

9, where n is the plasma refractive index (co11ision1ess

case) , W plasma frequency, a) frequency of incident laser P light and E is defined in the work of Hora.

In this case one has to take co as a function of P position, i.e., the case of varying electron density. As 47) shown by Hora this causes a deconfining acceleration of the

plasma which may be used for achieving higher compression.

Calculations show that this mechanism could be more efficient

than the scheme of Nuckolls.

One may therefore conclude that with the attainment

of high compression there exists good chance that the laser

driven fusion programme could become successful for energy

production and with the exotic reactions one might then over

come the radioactive hazards of fission processes in an

entirely novel way. 101

CHAPTER V

HEAVY ION PENETRATION:

In recent years much attention has been directed towards producing inertial fusion by injection of heavy ions from external sources. The possibility of bombarding solid targets by high energy electron beams to produce fusion reactions has been suggested long time ago by 4 8) Winterberg . There has been various theoretical proposals regarding the external sources for the heavy ions. One has been to use the accelerators for this, the technology for which has been very well advanced at present. However, due to substantial cost involved people have tried to modify this concept. We refer to the discussions in the litera- 49) ture in this context . Mention should be made here to another possible source which may make use of the proper ties of laser-plasma interaction. It has been experiment ally observed that when solid targets are irradiated with high intensity laser, emissions of not only electrons but also heavy ions take place which have the energy in MeV region or even higher. Siegrist, Luther-Davis and Hughes 50) have observed the emission of MeV ions from gold target 102

15 -2 irradiated up to 2 x 10 W cm laser intensity.

Theoretical explanation which has been proposed for the origin of the energetic heavy ions is that in the plasma which is formed from the solid the laser beam gets focussed to a region which has a very high energy concentration.

Ions getting there would absorb energy from the laser and get accelerated. The work of Kane and Hora^^ is of importance in this connection. Siegrist et al have found good agreement of their observed data with the self-focussing effect. One may subsequently use these ions to inject the plasma. For the high density pellet fusion people have suggested to use from 2 to as many as 100 beams with the peak power of 10 Megajoules in 10 nanosecs. The range of the heavy ions in high density plasma is of importance for this purpose. In the following we give the range of

Uranium in DT and HB^ plasma for various densities and energies - these are calculated on the basis of collective model. In fact the energy and the species of the heavy ions vary over a wide range for various proposals advanced, 6Q + e.g., there exists possibility of 12 GeV U ions production. 103

> H EV

Temperature

O LjJ

OJ fO P lasm a O)

sujo uj a&UD^j

Fig.25 104

tO EV

Temperature

oj Ll J ra > P iosm a o lu

O OJ D O

SUJO Uj 36UD£j

Fig.26 of 1 BeV Range in DT- PLASMA SOLID STATE ELECTRON DENSITY Fig. souo 27

96 ud £) 105 106

Q o

o: >

_J o

CD CO

- CM

siuo uj a&uDy

Fig.28 107

in EV

Temperature

Plasma »*- O

0) M O D O

sluo uj a6uo^j

Fig.29 o^o o f 5 8 e Vin MB Plasm a 2 c8 s SOLID STATE DENSITY OLLECTIVE MODEL Fig. SUUO 30

06uoy U) 108

Temperature in EV 109

CONCLUSION

The main result found in the above work is that the reheat mechanism predicts very optimistic fusion efficiency for thermonuclear reactions in inertially confined high density plasma. One has to attain initial densities higher than the solid state for this. The reaction of hydrogen-boron (11) which is completely clean from radioactivity could be a good possibility for energy production inspite of its low reaction cross-section.

Obviously one can search for other exotic reactions and apply the above calculations to see whether there exists a better choice of fuel. The calculations have shown that there could be some sort of ignition -- or self- sustained reaction mechanism -- at temperature of a few keV for the case of densities higher than the solid state.

Actually the numerical values of initial volume, input energy etc. could be used for the design of the fuel in order to plan the experiments.

There are two points left untouched in the above work and could be suggested for future investigation. One is the relation of the Fokker-Planck formalism to the Bethe- 110

Bloch theory in order to ascertain the reasons of the discrepancies for the range values obtained. This could be of interest as a general knowledge in plasma physics.

The second point of interest is the question of the effect of Bremstrahlung in the above calculations. It could be possible to take this into account by modifying the funda mental equations of thermokinetic expansions in the same way as the incorporation of reheat. Although this will give less gains the general features are unlikely to alter since the temperature of a few keV is not high enough to cause a drastic change. Ill

Appendix I

We calculate the integral

Vv'> - .» , dv' for the Maxwell-Bo 1tzmann v-v' distribution

3/2 v 2 f (v) e’Yv with y e 2 KT

+ 00 , 2 3/2 e‘Yv Then 7^77v-v dv

We note that

co ik .(v — v T ) dk as f o1lows v-v 2 7T k2

Taking axis in k-space along |v-v | for azimuthal angle

4> (0 to 2tt) and 0 being the angle for any other direct ion with respect to the axis one has

dk k dk s in0 d0 d(f> (k = |k|>

Then

I t I ■ • r pik.(v-v’) ^ ik v-v cos 0 f 2 ------^----- dk = k dk sin 0 d 0 dc{> J • vz J

I -> , I | -v ikv-v -ikv-v e 1 1 -e 2tt ik|v-v'

4tt sin x d x v-v

2tt « v-v Using this one has

4- oo 2 + 00 -f- a ik. |v-v' | ,2 r e-Yv f ■+, [ - y v ------dv’ dv |v-v'I 2 TT . J J J .

+ + 00 ■> ->■ dk ik . v -ik . v ,2 e -yv ’ 2 TT 7e

r 3/2 ■ r -¥ -y -> l TT dk ik . v 2 TT 2 Y J . . . k2 e

3/2 TT dk . n . -k /4y sin (kv) e l Y J

3/2 TT - Erf(/yv) l Y

Er f ( /yv) . v 113

Appendix II

Here we calculate the integral

+0° e Y |v-v’|dv

Substituting

■ -> ->• -> -> ’ = V + U : dv’ = du one can write

+ oo 2 2 -y(u +v +2(u.v)) u e du

2 2 -y (u +v +2uv cos 0) 2tt u 3 dun sin 0 d0 e

where in u-space one takes axis parallel to v to measure the polar angle 0. One has then

1 - ^ (Ii - V where

2 -(yu2 - 2yuv + yv2), u e du

and

2 -(yu2 + 2yuv + yv2) . u e du

52) Now one has to utilise the relation

b -ac 00 e-(aX +2bX+c) f(x)dx . i_ e a ^ / U e-y fi . b dy v*a & 0 b//a where a > 0 . Then

and 2 \ JL_ X_ _ /f

On further simplification this gives

4yv I dy + 2 1 Sy

= Z eV+ K Erf(/7v) . (,2+^) • 115

APPENDIX III

Here we discuss the scattering cross-section of an electron in the field of a heavy charged particle with the emission of a photon. The result which we have used is an approximate one known as the soft photon case. The general case of Bremstrahlung given by the Bethe-Heitler formula is more complex. The theoretical point of interest is that one does not need the cut-off given by the Coulomb logarithm for small angle scattering in this case. We assume all the notations of Bjorken and Drell — the units are such that

-ft = c = 1 and the fine structure constant

2 e

Now the scattering cross-section is calculated from the

S-matrix element for the particular process. This matrix element can be written down with help of Feynman rules as developed in quantum electrodynamics. In our case the matrix element Sf . between the initial state Ii> and the final state f> is

_2pJ -J/2 27T6(Ef + k-E.) f > i E,E . + 2 f ,i /2k f l i q i

where 116

f,i = ^(pf.sf)[(-i«!) {-iV + (-1V?pji^(-ii0]u(>1i>si)

This is same as (7.57) in Bjorken and Drell p. 122.

We are going to explain only certain steps in the calculation so that the final result is transparent. We note that in the above u(p,s) denotes a positive energy Dirac spinor,

V the quantisation volume, k the momentum of the photon emitted and q the momentum transfer. Now in the soft photon limit one can write after algebraic manipulation using the properties of Dirac matrices.

e .pf e .p f , i -iu(Pf’sf> ^0 usi>

The density of final states is given by a factor

V2d3k d 3 p f

( 2 TT > 6

The scattering cross-section will then be obtained by dividing the square of ^ by the flux which is a factor v_^/V, v^ being the velocity in the initial state and 2tt6(0) in order to obtain the transition rate per unit time:

d 3 kd 3 p 22 6 2 e.p e.p. ? luy ul da Z e m r------^)z --- 2-7- 2TT6(E-+.k-E.) 2kv.EE. k.Pf k.p. f l f l ( 2 TT ) 6

Pf dpf E dE f 117

Also utilising the properties of the positive energy Dirac

spinor one has for the elastic scattering cross-section

422a2m2 ,- ,2 (—) Kdft 'e 7,4 " lu Y0 ul lq 1

This is relation (7.11) in Bjorken and Drell. One can

therefore write for the scattering cross-section with photon

emission as

e . p da e-p - k2 dft. dk ( i>2 dft ^ Mft 'e J J k k . p k. p 2 k ( 2 tt )

In order to sum over the polarisations of the emitted photon

one notes that the expression

£ . pf £.p± k . pf k.pi

has the form L £ where L k =0. Then the polarisation d d d d i sum becomes simply L L — sew Mandl J for this. One then

obtains

2 da da a 2Pfpi m_____ tt f kdk / dft dft £ dft.) / k k . pf k.p± f e 4tt (k.pf) (k . pJ 2

This has been worked out in Bjorken and Drell and in the

nonre1ativistic limit has the form 118

dg / do N 8a 1 max n2 . 2Q /0 d^ ^ (77T")dM,. e« 3 7T ln ik------. p Sln 0/2 r mm

B deno ting the electron velocity. 119

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Note on adiabatic expansion

For an ordinary reader it would seem questionable to apply the concept of work done by pressure during free expansion. For example, in the Joule-Thoms on experiment although the gas expands, there is no temperature change "after" the expansion has come to an end. This is a change from one equilibrium state to another. But in the intermediate states of expansion there is work done by pressure, and this is the case we are dealing with here.

Note on equation (4.4)

The use of equation (4.4) may need a comment due to its resemblance with the perfect gas law. We recall that the origin of the concept of pressure is the collision mechanism. For a Fermi-Dirac gas the law

P = fep holds true where e is the energy per molecule and p the density. For a proof of this see J.E.Mayer, Encyclopedia

II of Physics, ed. S.Flugge,Vo1ume XII,Springer-Verlag (Berlin),

1958 ,p.181. Thus one simply has to consider T as given by the average energy per particle in plasma with a numerical factor. 125

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GH 14 Reaction Gains of Hydrogen and Boron (11) at Laser Compressed Plasmas. H. HORA and P . S . RAY, Dpt.Theoretical Pliysics/School Phys . U.New South Wales, Kensingt-On,N.S.W. Australia ■■Numerical calculations of the reaction gains for compressed hydrogen-boron (11) plasmas at inertial confinement are performed including the inaccuracy of the cross section for the nuclear reactions. For the pessimistic case of neglected alpha particle reheat, the fusion gain for laser induced compression is derived by a similar formula as was reached before for deuterium or deuterium-tritium. This formula is in agreement with the criterions derived by Kidder^, to substitute the static Lawson- criterion for the dynamical case of inertial confinement. Though the break-even energies for hydrogen-boron are relatively high, compression by nonlinear force driven fast imploding cold plasma shells, meets reasonable design parameters.

~~1r. E. Kidder, Nucl. Fusion 14 , 79 7 ( in"''~~

~~Bull. Am. Phys. S o c. , 21 , 73 , ( 1 97 6) 130 Penetration length of alpha particles at laser-produced thermonuclear reactions1 By P. S. Ray and hi. Hora~~

~~Deportment of Theoretical Physics/School of Physics, University of New South Wales, Kensington, 2033, Australia~~

Abstract Following Bagge's correction of the Bethe-Bloch-formula for plasma layers by the non-linear radiation force [10] to arrive the penetration of relativistic electrons in hot and dense at ideal conditions of adiabatic compression and increased plasmas, we derived an analytical formula for the penetra efficiencies of thousand times less imput energy for the same tion of alpha particles in nuclear reaction plasmas. The es reaction yields [11] overcoming the upper limit of the laser sential parameter is the limitation of the radius for the inte intensity of the gasdynamic compression scheme [12]. The gration of the Coulomb collisions, which is given by the Debye importance of the u-particle reheat was demonstrated in length. A representative evaluation is given for the hydrogen- numerical examples by Goldman [13]. The correction of the boron-11 reaction. The resulting length may influence nuclear u-particle reheat due to the penetration length in relation to reaction yields of nearly solid state density, indicating a pos the diameter of the reacting plasma, has not yet been in sible explanation for the discrepancies in measured and cal cluded in the published numerical work of laser produced culated neutron gains form laser-produced plasmas. A further plasma, however, an analogous example exists for the rela influence for self-sustained reaction fronts is expected. For tion of the fusion reaction length and the plasma size, where densities exceeding 104 times solid state densities, no influ a certain correction of the nuclear reaction yields could be ence is expected. shown [14], In the theory of the self-sustained nuclear combustion front, Zusammenfassung the basic derivations usually neglect the penetration of the Reichweite von Alphateijchcn in lasererzeugten thermonuklearen Reaktionen produced u-particles [15] for reasons of simplification, how Aufbauend auf Bagge s Korrektur der Bethe-Bloch-Formel fur die Reichweite ever, Zeldoich and Raizer [16] have included the penetration relativistischer Elektronen in heiBen Plasmen wurde eine analytische Formel fur die Reichweite von Alphateilchen in Kernreaklionsplasmen hergeleitet. based on the Bethe-Bloch-formula, while the correction by Der wesentliche Parameter ist die Begrenzung des IntegrationsradiuS auf the collective effects of the plasma was not taken into ac CoulombsloBe, die durch die Debyelange gegeben ist. Als Beispiel wird count [15]. The very transparent model of a stable supersonic die Wasserstoff-Bor-11-Reaktion ausgewertet. Die erhaltenen Ldngen durf- nuclear combustion front of Ahlborn [17] includes the energy ten die Kernreaktionsausbeuten nahe Festkorperdichte beeinflussen, was eine mogliche Erkldrung fur Diskrepanzen in gemessenen und berechneten of the generated u-particles. Neutronenausbeuten in lasererzeugten Plasmen ist. Weiterhin so!Ite der EinfluB auf selbsterhallende Reaktionsfronlen bestehen. Fur Dichten fiber Bagge's correction of the Bethe-Bloch-formula 10'facher Festkorperdichte wird kein EinfluB erwartef. For fast (relativistic) electrons, the decrease of the energy E

INIS DESCRIPTORS on the penetration length x in unionized material of an elec ALPHA PARTICLES HOT PLASMA tron density N0 is given by the Bethe-Bloch-formula [1; 2]. LASER-PRODUCED PLASMA RANGE 2ne‘* N 2m E v2 PLASMA DENSITY THERMONUCLEAR dE ______e e m log (- ( ) DEBYE LENGTH REACTIONS dx )- 1 m v2 •E e BB<1_v2/c2)

Introduction where c is the velocity of light, v the velocity of the electrons The reheat by the generated u-particles in laser-produced and Em is the maximum energy transferable from an atom plasmas influences the net energy gain of the reactions es to fast electrons, e is the charge and m,, the mass of the elec sentially. Also in laser-ignited self-sustained nuclear reaction trons, and the "mean binding energy" is fronts, the reheat by the generated u-particles is important. E,m ~ 15eV (2) While the penetration length of alphas is very well known in solids, liquids or in neutral gases and can be described by Bagge substituted this energy for the penetration of a fully the Bethe-Bloch-formula [1; 2], the knowledge for plasmas of ionized plasma by the electrostatic energy of the Debye high density (solid state density and higher) is very poor [3] sphere of a radius r() also taking into account the indirect knowledge from large Eb = eVrD (3) nuclear reactions. Laser-produced plasmas [4] and plasmas Though the influence of this change is only due to a logarith of solid state density generated by relativistic electron beams mic function in Eq. (1), a remarkable change for the condi [5] give a first opportunity to study the penetration of fast tions of cosmic radiation had been shown [6] to explain some particles in high-density plasmas under laboratory conditions. anomalies. The success of Bagge's generalization of the Bethe-Bloch- formula [6] for fast electrons describing the drastic decrease In terrestrial experiments, a strong change of the penetra of the penetration [7] in full agreement with measurements tion length of 2 MeV electrons in a CD2-target at high elec [8], was the motivation for this work to study the similar tron currents was measured. For solid material, the length change for alphas. is 13 mm, while the measurement of the neutrons from nuclear fusion reactions indicated an increase on varying thickness The importance of the u-particle reheat for laser-compressed with a saturation at 3 mm [8], It can be assumed that the nuclear reactions is evident and was included from the be target was immediately changed into plasma by the electron ginning in the computations for compression by gasdynam- beam. ically driven ablation [9], An improvement over the gasdynamic ablation is possible by using the acceleration of thick While several numerical simulations have tried to show the decrease of the penetration depth of the electrons compared 1 Dedicated to Professor N. Riehl on the occasion of his 75th birthday. with the Bethe-Bloch-formula (1) by assuming a return of the

Atomkernenergie (ATKE) Bd. 28 (1976) Lfg. 3 155 131 electrons or by implicitely described collective effects, a clear Ax . Thus the energy loss by the heavy particle, which is the analytical explanation of the reduction of the penetration total energy transferred to the electrons, is length was given by the use of Bagge's correction in the Bethe-Bloch-formula [7], 4 tt Z2 e1 The importance of the reduced penetration of relativistic -AE N bdbAx e electrons in high density plasma is given by its drastic im b2 v2m provement of the expectable nuclear reaction yields follow min ing the concept of Winterberg and others [5; 18; 19] where max finally the concept of compression, as known from the laser 4n Z2 e4 N( db interaction [4; 9], was applicable at the electron beam inter -AE v2m b action [7; 20]. min Here blliav and b||ljM denote the maximum and minimum val Alpha particle penetration ues of the impact parameters. One therefore has the follow The derivation of a penetration formula of the kind of Eq. (1) ing expression for the rate of change energy by the heavy for alphas in plasma is easier than for electrons because the particle: alpha-energies of up to lOMeV are non-relativistic and the dE . „ Z. 2 e„ ■» b____ - -J— = 4 TT N log (10) deflection of the colliding fast alpha particles is less than dx e m v2 min that of electrons. The range of the alpha particle starting e from the point in the plasma where it is produced can be For bmill there exists already the classical estimation [1; 2] obtained in close analogy of the theory of alpha particle Ze2 penetration in matter as done in nuclear physics [21]. The ^min^ cl (11) m v2 energy loss by it is mainly due to the inelastic collisions with e the electrons in plasma. One may assume that the motion of We shall assume this value. For bllinx there is a natural char the alpha particle is undeflected due to the interaction be acteristic in the plasma state. As the positively charged alpha cause of its heavier mass-the very low probability of col moves through the plasma it acquires a cloud of negative lisions with nuclei (as seen immediately from Wilson chamber electrons until the positive charge is completely screened. pictures) is not changed by the ionisation of the atoms in the This is the Debye sphere. One can assume the electrons plasma. situated beyond the Debye sphere do not absorb energy Consider, as in Fig. 1, the interaction of a heavy charged from the incident alpha. Thus particle Zc moving in the x-direction with an electron e. From the symmetry of the problem the net effect on the electron is bmax “ *d that it experiences a force in the direction perpendicular to where 7.p, the Debye radius, is given by that of the path of the charged particle. The corresponding force is (13) /4tin e2 e — cos 9 = Ze2 — (4) r„ 2 r 3 T is the plasma temperature and k the Boltzmann constant. Thus the energy loss per unit path of the heavy particle is If it acts for a time interval At the gain in momentum is kTm dE Z2e Ne log ( (14) A p = Ze2 — At = —e- — Ax (5) dx 4ttN e6 Z2 r3 vr3 e where v is the velocity of the heavy particle (Af/Ax = 1/v) in analogy to the Bethe-Bloch-formula (1), however with the and Ax is the distance traversed by it. The total momentum correction for plasma, if mn denotes the mass of the alpha Ap gained by the electron is particle and E its instantaneous energy

(Ap) 2E E total m 2m. (15) r 4*oo Ze b dx Ze2 b 2ze3 when m., is the proton mass. bv (b2+x2)3/: Thus m , kTm 2E S (E) = ~ = 16Tie4 -£ N i log ( e Thus, if AE denotes the total energy gained dx m e E ^ -) (16) 64-rrm 2N e6 P e total 2Z2 e4 AE 2m (7) The range R is then obtained as e b2 v2m e o where in,, denotes the mass of the electron. EdE (17) Let N(, denote the plasma electron density. Then the number S (E) of electrons situated per length Ax of the heavy particle path 16ne log(XE2) between b and b + db impact parameter value is 2;tNcbdb where kTm (18) 64nm 2 P and where E„ is the initial energy of the alphas. In the follow Fig. 1: Collision of a particle of charge Ze moving along x colliding an ing we use the value En — 2.894 MeV which is the initial en electron e at a distance r. The impact parameter is b defining the angle ergy of the alpha particle produced by the reaction [21] of

156 Atomkernenergie (ATKE) Bd. 28 (1976) Lfg. 3 B 11 (p, <<) 2(< which is of special importance for laser com 132 pression [II]. The alphas for the DT or DD reaction are of similar values [4], Using the function of the integral logarithm

~~Ei (x) dt, (19) one gets 2 m (2C) 2 h s2 Ei (1°9 >~~

~~One criterion to be satisfied is that~~

b >> b . max min (21) which is equivalent to Fig. 3: Penetration length FI for 2.89-MeV alphas in HB plasma af varying densities N(. kT >> 2.413 x 10"25 Ne (22) minor compression of less than one hundred times the solid This restriction is not severe for plasmas of solid stale den state density N, . Following the examples [11] of a reaction sity (solid HB11 has Nr — 2.82X 1 CP cm•'*), however for the gain of 80 for DT with r0 = 325 joules and a compression up interesting densities of up 10’’ times of the solid state density to a density of Nmax 10'Ns, the minimum radius of [11], restriction (22) to higher temperatures is of importance. 2.1x10 'em is very much larger than the corresponding Results penetration length of the alphas. The same can be seen for HB 11 with =3 2xlOfi joules, Nmnx — 10’’ and a minimum The evaluation of the penetration length I? of Eq' (20) had to radius of 5.2x10 'em. Any influence of our result can only use the computer routine for the integral logarithm. In Fig. 2 be in peripheric areas of the highly compressed plasmas. the penetration length R is given for the alphas of the HB 11 reaction [22] for plasma of solid state density in dependence The correction of fusion gains for medium densities may be on the plasma temperature T. The decrease of the penetra- a critical factor for understanding the discrepancy of meas ured and calculated reaction gains of the present laser- fusion reactions [4]. For the self-sustained nuclear reaction fronts, where the den sities are not far away from the solid state [15-17] a change of the present results due to the corrected penetration length of the alphas in hot and dense plasmas (Eq. (20)) should be of influence. (Received on 7. 5. 1976)

References [1] Bloch, F.: Ann. d. Phys. 16 (1933) 285 -Tj------y — ry- ...... — s 10 10 10 10 10 [2] Bethe, H. A.: Ann. d. Phys. 5 (1930) 325 plasma temperature (3| Sir Ernest TiHerton (private communication) kT (electro^-Volts) [4] Hora, H.: Laser Plasmas and Nuclear Energy. New York: Plenum 1975 Fig. 2: Penetralion length /l for 2.89-MeV alphas in HB plasma of solid [5] Freeman, 8 , H. Sahlin, J. Luce, O. Zucker, T. Crites, C. Nelson: Bull. stale density of varying temperatures Am. Phys. Soc. 17 (1972) 1030 [6] Bagge, E. R.: 13th Int. Cosmic Ray Conf., Denver, Aug. 1973; The Origin of Cosmic Radiation. Munich: Thiemig 1968 tion length R at higher temperatures is partially due to the [7] Bagge, E. R., H. Hora: Atomkernenergie 24 (1974) 143 fact that the end velocity of the alphas which is given by the [8] Kerns, J. R., C. W. Rogers, J. G. Clark: Bull. Am. Phys. Soc. 17 (1972) temperature of the plasma, is reached earlier than at lower 690 [9] Nuckolls, J., A. Thiessen, L. Wood, G. Zimmermann: Nature 239 (1972) temperatures. 139 The penetration of alphas in solid non-ionized HB is [10] Hora, H.: Phys. Fluids 12 (1969) 182 4.6x10 cm, which is of remarkable difference to the cases [11] Hora, H.: Atomkernenergie 24 (1974) 187; Kvanfovaja Elektronika (Febr. 1976) of the plasma. [12] Hora, H.: University of New South Wales, School of Physics, Rpt. 25 Fig. 3 shows the penetration length for the alphas of the (1975) HB11 reaction for plasmas of various densities. For high [13] Hora, H., E. Goldman, M. Lubin-. Univ. Rochester, LLE, Rpt. 21 (1974); Goldman, E.: Plasma Physics 15 (1972) 289 densities, the restriction of Eq. (23) is becoming effective and [14] Hora, H., D. Pfirsch: Laser Interaction and Related Plasma Phenomena the plots are ending at temperatures T0 which are ten times (H. Schwarz and H. Hora, eds.) New York: Plenum 1972, Vol. 2, p. 515 the lower limits Tmill of Eq. (22) [15] Bobin, J. L.: ibid. Vol. 3B (1974) p. 456 [16] Zeldovich, Ya. B., Yu. P. Raizer: Physics of Shock Waves and High Temperature Hydrodynamic Phenomena. New York: Academic Press, To - 10 Vn - 2-413 - 10-“ (23) Vol. 1 & 2 (1966/67) This limitation comes close to critical condition at the inter [17] Ahlborn, B.: Phys. Letters 37A (1971) 227 esting densities of 10' to 10’’ times of the solid state density [18] Yonas, G.: Lecture presented at the "Ellorc Majorana" Conference Erice, June, 1973 at temperatures close to 10’ eV for the HB 11 reaction [11; 23]. [19] Winterberg, F.: Nucl. Fusion 12 (1972) 353; Z. Nalurforsch. 30A (1975) For the DT and DD the conditions are less critical [11]. 976; L. I. Rudakov: Sov. Phys. JETP 32 (1971) 1134; Pulnam: Lecture presented at the "Ellore Majorana'' Conference, Erice, June 1973 Conclusions [20] Clauser, M.: Bull. Am. Phys. Soc. 20 (1975) 1301 Penetration length comparable to that of the size of the [21] Fermi, E.: Nuclear Physics. U Chicago: Chicago 1951 [22] Oliphant, M. L. E., Lord Rutherford: Proc. Roy. Soc. A 141, (1933) 259. plasma geometry are expectable only for reactions with low T. Weaver and L. Wood, Lawrence Livermore Dept. UCID-16230 (1973) initial energies ,<0 of the plasma of less than 1 keV and at a [23] Hora, H., P. S. Ray: Bull. Am. Phys. Soc. 21 (1976) 73

~~Atomkernenergie (ATKE) Bd. 28 (1976) Lfg. 3 157 133 NUCLEAR FUSION 1C R (1970) - LETTERS~~

ON TIIE RANGE OF ALPHA - PARTICLES IN where m(y is the mass of the alpha, V the instan LAS Ell-PRODUCED SUPERDENSE FUSION PLASMA taneous velocity of the alpha, kn,ax is a cut-off factor due to small distances, and J is defined in P.S. RAY, II. HORA (Dept, of 'I lieoretjc.il Physics, The Ref.[3]. The reciprocal of kmaY is a distance below University of New South Wales, Kensington, N.S.W., which binary collisions are predominant. The Australia) integral being finite, we see that for kmax ~ l/XD, (dE/dx)u ~ 0. Gasiorowicz et al. [3] have utilized k ~ l/d, where d is the interparticle distance. Cohen et al. [4] have argued for the choice of la view of the possibility of energy production k ~ 1 /A ,-j. For the case of superdense plasma from laser-induced nuclear fusion reactions in this choice should be good enough. I hus v. e have deuterium-tritium or other pellets in plasma state dE/dx~~: (dE/'dx)N. of solid-state electron density or even of higher Now, the rate of energy loss 6(E) of a heavy density [ 1), it is of importance, for various pur particle per unit path due to binary Coulomb colli poses, to know the penetration length of alpha- sion is given by [ 5 ] particles released by fusion reactions in such superdense plasma. For example, the absorption b of such alpha-particles in the adiabatically dE 2e4 max S(E) ■= 47rNe (1) dx b . expanding plasma could raise the temperature, ev2 rr.in thereby influencing the reaction gain, e.g. as reported for the hydrogen-boron fusion [2]. Though where Ne is the electron density, Ze is the charge the penetration of alphas in plasma has been of the heavy particle (Z = 2 for alpha), V its instan discussed by several authors, it seems to be of taneous velocity and m^, is the eleciron mass. Here importance to treat this problem for extremely binax and bmin are the maximum and minimum high densities to such an extent that numerical values of the impact parameter. As explained values are available. To estimate these values, above, we can set b = AU, anc^ hn.in the classi we note that the inital energy of the alpha when cal value (bmin)cl - Ze2/meV2. This gives for S(E) produced in plasma — which is 2.894 MeV for the reaction uB(p, 2a) a or a similar value for the D-D or D-T case — is substantially higher than the S(E) 16:re4 —2. N llnf kTmK .1 (2) m e E g 64rm£ Ncet_ electron energy of the plasma corresponding to e temperatures of the order of 1-100 keV. The alpha-particle, therefore, moves rectilinearly where kT is the plasma temperature and m? the from the point where it is produced in the plasma proton mass. The range is obtained from until it loses its energy by collisions with the electrons, down to the magnitude of the thermal Eo 1 _1_ r EdE energy. This defines its range. The rate of energy (3) loss per unit path can be written as 16ffe4 Ne mp J in (rE2)

dE _ ( dE\ / dE\ where d* "\dx/N +Vdx/R - kTn^ 1__ L r 64 m'f; N e6 where (dE/dx)N is the rate of energy loss of the F e alpha due to its reaction with its "nearest neigh bour" and (dE/dx)R the corresponding value for Here E0 is the inital energy of the alpha particle. the "rest" of the plasma medium. To define the nearest neighbour we utilize the Debye sphere of Introducing the function radius XD about the alpha within which individual x binary collisions with electrons under the Coulomb Ei (x) = f — dt law take place. For the rest of the plasma beyond J t the Debye sphere, the plasma acts as a continuous medium of high electron density. The electrons we obtain in this region experience forces not only from the alpha but also due to perturbation of other electrons caused by the presence of the alpha. Thus the rest P2 m _ li=2kT of the plasma acts back on the alpha as if it were ^e Ei(ln K» a polarized medium. The effect of this has been established by Gasiorowicz et al. [3]. Using their One restriction to be satisfied is that b » b max min ' result for this force in the direction of motion of which means the alpha, we obtain _o c. kT» 2.413 X 10 N, (4) +1 ( dE\ — In (k2 A2) / dp p J(p,v) where kT is in electron volts and Xe the number of WR U1 0 max D J -1 electrons per cm*3.

~~535 134 NUC.IFAR FLSION 1>' 3 (1970) - LETTERS~~

Wc note that Jackson [6] has given the expression which is similar to (2), where k() = 1 /A D and is the plasma frequency. However, it is valid only for impact parameters b Ay and the argument dL (Ze)C , / 1.12 3 knV\ = ------— w 111 1 ------L?- ) of the logarithmic function is different. It does not dx v4 ? V UP / contain the electron density N . So it does not allow one to see this effect of the high electron density — 10-1 although the function In is not much susceptible to I a change of the argument, the function Ei is very Ne = 10 times solid state susceptible to the contrary. Figure 1 shows the result of the numerically evaluated ranges R of alpha par ticles in plasmas in dependence on the plasma temperature for various ion densities of HuB-plasma.

~~REFERENCES~~

[1] HORA, H.. Laser Plasmas and Nuclear Energy. Plenum New York, 1975. tf to io2 101 io‘ ios t) [21 HORA, H.. RAY, P.S., Bull. Amer. Phys. Soc. 2J_(197G) 73. kT- PLASMA TEMPERATURE [3] GASIOROW1CZ, S., NEUMAN. M., RIDDELL. R.J.. Phys. Rev. 101. (electron-volts) (1956) 922. FIG.l. Penetration length (range) of 2.89-MeV alphas in HllB plasma [4] COHEN, R.. SPITZER, L.. McROUTLY, P.. Phys. Rev. 80 (1950) 230. versus plasma temperature for various ion densities. The curves end at 15J FERMI, E., Nuclear Physics. The University of Chicago Press (1951). the left, because of condition (4). [6] JACKSON. J.D., Classical Electrodynamics. p.45l, Wiley (1965).

~~(Manuscript received 19 January 1976 Final version received 26 April 1976)~~

CONDITION FOR ABSOLUTE CONFINEMENT OF For practical reasons we assume that the fusion ALRIIA-PARTICLES IN AXISYMMETRIC TORI reaction only occurs on the elliptic magnetic axis with maximum plasma pressure. (Fig.l). K. IKI TA (Institute of Plasma Physics. Nagoya University, For simplicity, we consider the axisymmetric Nagoya, Japan) toroidal plasma given by the following magnetic- flux function:

The orbit confinement of fusion-generated alpha- particles is one of the key methods for confinement devices to be operated as steady fusion reactors. In nuclear reactions of interest in connection with --££(1-kV+*V/2|. rUzW (1) fusion reactors, the generated alpha-particies have energies of the order of 1 MeV. The ques tion then arises whether the assumption of alpha- where the plasma is confined in a spherical particles having the same confinement time as the region (i.e. r2+z~SR~) and the plasma-vacuum fuel ions [ 1 ] is justified or not in reactor studies interlace is assumed to be a sphere of radius R [2), since the time of transit of the alphas across the where a cylindrical co-ordinate system is used toroidal field due to VB-drift is much shorter than and the toroidal field coil is placed along the z-axis their slowing-down time in standard tokamak (Fig.l). plasma. Moreover, we are concerned with the In this model, we have one elliptic magnetic problem of whether a single alpha-particle having axis at an energy of the order of 1 MeV can be trapped or nut in the collision-free plasma of a magnetic r = R/n/I" and z = 0 (2) device of pi'actical interest. The purpose of this letter is to derive a necessary Around the elliptic magnetic axis, the cross- condition for an axisymmetric torus to act as an sections of the magnetic surfaces form a family of absolute trap for fusion-generated alpha-particles. ellipses. Thus, the magnetic axis described by

~~536 135~~

~~Bull.Am. Phys. Soc. , 21 , 1 153 ( 1976)~~

8 A- 3 Thfirroalization Length_9f Alphas and. other Nuclear .Particles in High Temperature Plasma. P. S. RAY and H. 1IORA, Dept. Theor. Phys., UNSW, Kensington-Sydney, Australia. - The penetrat ion length R of alphas, protons or other particles from nuclear react ions in high density plasmas is essential for the reaction yields in laser compressed plasmas as well as in tokamaks, high energy particle beam interaction with dense plasmas and other reactions. The often used formula R ~ T3 n gets a remarkable correction for T between 101 and 103 eV, if the thermalization is described by a Fokker-Planck equation taking into account the elastic as well as inelastic scattering. The emissions of soft photons during interaction with plasma electrons causes a theoretical correction to the usual Fokker-Planck formalism and is based on the scattering theory as developed by Schwinger.1 ’One basic quantity is the lower limit of the photon energies emitted for which the Bagge frequency from the generalization of the Bethe-Bloch formular for plasmas is used ns it was successful for calculating measured penetration depths of relativistic electrons in plasmas of solid state density. The range is susceptible to the initial energy as well as the mass of the heavy particle, as is illustrated by numerical examples.

~~11J. Schwinger, Phys. Rev. J6. 790 (1949).~~

~~1 136~~

On Thermalisation of Energetic Charged Particles in Fusion Plasma with Quantum Electrodynamic Considerations P. S. Ray and H. flora Department of Theoretical Physics The University of New South Wales, Kensington, New Sout Wales, Australia 2033

~~(Z. Naturforsch. 32 a, 538 — 543 [1977]; received March 21, 1977)~~

The process of energy loss of charged heavy particles in plasma has been discussed taking into consideration photon emissions by electrons of plasma during scattering as is necessitated by quan tum electrodynamics. Although the range in general increases with temperature, it is found that the numerical formula of former treatment giving a range as T3/2 is not in general, obeyed. The range is susceptible to the initial energy as v eil as mass of the heavy particle as can be seen from the cases of alphas and reaction protons for DT, DD (-*-p + He3) and D He3 plasmas.

1. Introduction Winterberg’s formula was used for the calculation of the nuclear fusion yields of inertial confinement The penetration depth of alphas, protons or other of laser produced plasmas c and in the general codes MeV-particles from nuclear reactions within thermo for laser compression by Brueckner and Jorna7. nuclear plasmas is of importance for the reheat and Though Brueckner himself was aware of the much total energy gain of these plasmas. In the case of more general properties of the penetration depth8. tokamaks, the theory for deriving a more precise As has been emphasized by Feynman 9 it is im expression of the penetration depth has been elabo possible to scatter an electron with the emission of rated by Diichs and Pfirsch 1, where a remarkable no photons. The real photon emissions in scattering improvement for the reactor conditions of tokamaks brings with it the problem of infrared divergences can be expected. The penetration depth plays an of quantum electrodynamics and necessitates a cut essential role for all concepts of fast-particle driven off factor and for this one has a natural estimate in heating and (eventually compression) of plasmas of plasma physics. One therefore has to reconstruct the solid state density or higher. The decrease of the Fokker-Planck formalism with this aspect. This is penetration depth by a factor 14 or more for very done in this work and also the explicit formula for dense McV-clectrons in CO., plasma of solid state the range has been derived. We recall that the ener density compared to solid CO., material, was a first getic particles, e. g. alphas in DT-plasma have initial indication of the strong differences2 to the usually energies of some MeV, whereas this plasma tempera known behaviour of solids. The easy explanation of ture is of the order of keV. The motion of the heavy these facts 3 by using Bagge’s generalization 4 of the particle is therefore rectilinear until they lose energy Bethe-Bloch-formula, encouraged us to evaluate the by collision mainly with the electrons. The range more general cases of alphas and other heavy can then be defined as the distance travelled until charged particles in high density high temperature the energy is reduced to the thermal values. plasmas. An application to plasmas of the tokamak type will need more studies of additional forces, as It may be noted that in the work of Diichs and e. g. incorporation of Lorcntz-forces, etc. Pfirsch the Fokker-Planck formulation employed Usually, the reheat in laser produced plasmas is uses energy as a variable of the distribution func calculated on the basis of a formula for the penetra tion. A similar kinetic equation has been used by tion depth R derived by Winterberg 5 Tsui et a!.10 in their work on the time development of the energy distribution function of the alphas. In R~TW. (1) contrast with these treatments, we are not evaluating the values of the distribution function, but we use more general cross sections and calculate the slow ing down length R of the alphas from the Fokker- Reprint requests to Mrs. J. Stanic, Dept, of Theoretical Physics, University of New South Wales, Kensington, Planck coefficients explicitly for numerical evalua New South Wales, Australian 2033. tion. 137

~~P. S. Ray and II. Hora • Thermalisation of Energetic Charged Particles 539~~

~~2. The Fokker-Planck Formalism where~~

Consider a distribution of different species of 3ni o“) D(n) V = °ac (yi, vin) particles denoted by indices s, t, ... If fs[r,v,t) is 33u;,... 3t/jB * the distribution function for species s, its evolution is governed by the Fokker-Planck equation which If one chooses ys{v) = jj v2 — o 2 v»2 so that all 3=i we write in the form of Kramers the derivitives of y> higher than second order vanish, 3/s , 3/, Fi 3/, v ~ (-1)" ,, , one obtains: ■57 + v‘ a“ + — dviaT = 2t 71 2=» 1 ... ■ n\- £>'"> («(») /.) • t lr (n* “2<')+y ^ (n* v~ t’i

The a’s are the Fokker-Planck coefficients. One giving na = <5(r—ra) and ^ = Y>(ua). One should note that thermodynamic equilibrium is not will then obtain for the energy Fa=^mava2 after assumed for the validity of (2). We take for sim integration with respect to r and V: plicity only two species of particles are present e. g. dFa/df = ma (2 ^’a,i m a 2 ^ii) (6) in a fusion plasma the electrons (subscript e) and « i the alphas (subscript a). We shall neglect the inter for the rate of energy loss. action among the alphas themselves. We further To evaluate further one has to substitute the val neglect the interaction a-ions. To obtain the rate of ues of rii and as functions of Vi. Rosenbluth et energy loss of the alphas one has to go over to the al.11 have calculated these on the basis of Coulomb hydrodynamic equations. Let y>(v) be a function scattering and their result has been mostly utilised of velocity 1’ only. Its average y> with respect to the in plasma physics. Their calculation gives: distribution / is defined as V = f Y f dvff f dV = (\/n) f xpfdv (3) cii = r-^-h{v) , ai’i where d\ = diq di/\» di/3 and n = n(r,t)=ff du the (7) local density. Multiplying (2) by yj[v) and inte b,i=rdv7^9<'v) grating with respect to dv one obtains where ^ m.hO'k J_ 3 (yf,) (a) T ("a V(a)) + -£r ("a v Vi (a)) = dx-t ma a Vi (-1)" tf(l>) = /di//e(u') + 2-- , f V D{n) {alh fa) dP n = i n! and for T {z —2 for alpha) where averaging is taken with respect to the alpha (4rr e4 z2/ma2) lnyi (8) particle. Now by partial integration one has with A representing the Coulomb logarithm: / v D<-<(oft/.)dt>-(-!)>/<> 0<"> V/»d» . 3,t;.d 31 /nay i "(-l)-n, af;,/)<»)./*>. A“Tkrl' *]/[*-) In the absence of external forces one can therefore For the electrons we shall use the Maxwell-Bollz- write (4) mann distribution

~~I, (#, ■?'*>) + §3 - of-o 0<") vw I.W-Np-)*1 e-T* (9) 138~~

~~540 P. S. Ray and H. Hora • Thermalisation of Energetic Charged Particles where A'e is the electron density, h T plasma tem perature and y = me[2kT. One obtains then~~

~~{-2 6,,. —r7~g I 2 i dv,2 /Ve where 3/2 1 / = / dt/ e~rV ' Erliy^v) . Fig. 1. Feynman Diagram showing electron scattering in 17-17'| external field together with photon emission. Furthermore,~~

2 a; Vi -=rZvi £—h{V) Schwinger first calculated this contribution in a relativistic w’ay. It has been also calculated with the cl help of Feynman rules by Bjorken and Drell n. In r\i+ —)/ve(— V V mel \n the non relativistic limit the corresponding cross- One thus obtains (mA me) section is dE 4.T22 e4 2 rne v-\ rdo lnAAP (i?.l = (• d l yVr t2 kT 2 k T J \dU QjQJi \c where a is the fine structure constant (a = 1/137) Erf (10) 2 kT and /? = u/c i. e. the relative velocity, 0 the scattering angle, (do/df?)c is the Coulomb value. With respect Now the range is defined as the distance travelled to the photon frequency to emitted: dk/k — dtofto. until averaged thermal energy Fth = \ kT is reached. To find the contributing cross-section one has to Thus if F0 *s 'be initial energy of the heavy particle integrate over all possible frequencies emitted say the range R is given by between comjn and comax

~~r dE f vdE / do \ _ 2 a z2 e4 / ____1____~~

R "" jJ - (tXtjdx)(d£7di) ) dEjdl \d-Q)i 3 tic2 mV \co\ n,in/ u2sin2 0/2 * Et h This expression is divergent for comjn—>0 which is ma (k TV 1 f x3 dr the famous infrared divergence. We can use a cut (11) m(. \ e2 j /Ve In/f riz2 J f(x) off for this and choose the “Bagge” frequency where e2/XD = htoB fix) — Erf (x) where XB is the Debye radias: Xp = Yk T/4 ti Ne e2 . y?i Bagge4 first used this value for the binding energy Jp 0_ _g£e correction to the Bethe-Bloch formula for the cal i/fi' -vk T m. culation of penetration depth of relativistic electrons in plasma. He obtained thereby ranges R being of As the temperature k T increases we have x*—>o:1. better consistency for explaining cosmic radiation Thus the integral in (11) decreases to zero. How and resulting in a remarkable agreement 2 with the ever, the factor (kT)2 increases much faster giving measured strong decrease of ranges in high density a range which will increase with (kT). plasmas for relativistic electrons. For the maximum frequency emitted one can use the instantaneous 3. Quantum Electrodynamic Contribution energy of the alpha: E — h (omixx. One thus obtains:

~~As mentioned already there should be a theoreti do \ _ 2 a z2 e4 1 cal correction to this formula since there is always~~

If one uses the sum of the differential cross-section, a- f dv'fe{v'){Av} namely mP 16a z 1 e4 , . la Kfdv’ /,(»') -r£—^ m. 3 mP c~ \V—V Since \V — V \v-v there will be additional terms for a,’s and fc4-/s. We v-v' 0y now work these out in detail: From the definitions one can write ai = f dv'fe(v') {Av} , mn 16 a z~ e4 y \3/2 3/ a In KNe bij^fdv'fAv') {Ji’idvj}. 3 me2 c2 d) Here {Av} denotes the change per unit time in the where > velocity of the alpha by collision. Suppose V and V 7 = f dv' e~yv * ] v — v | V>{v) denote the velocity of the alpha and the electron V before collision and U their relative velocity W{v) = ~~~e rv* + Vn Erl (Vy v) ^u2+-“). u = i>-r', u = V+ ~u, v'- V — ~u, Thus M = -f me, MV — ma V + me v'. me 16are4, , u.v=------—In k Now the change in TL by collision, denoted All, can ma 3 mg c~ fc) be obtained considered Figure 2. Similarly i 2 bu — 2/d?//e(v') {Av(Avi}

~~A u m, 3 mP- c~ i\Ji1 giving 2 u bti ^ a-V .~~

Fig. 2. Change in it by coiiision, 0 denoting scattering angle. For the rate of energy less one therefore obtains dE 4 7i z- e4 . 1 —- =------/Velnyi — With notations as shown in the figure one has dt m„ v O j All | = 2 u sin , y.~i ikfv^ -~2kT\L “(1ft,-) 16 a z2 e4 1 1 An — \ All ci sm• —® + In KN, 3 \!7i me c~ v y Elf (fw f rne v2 | + cos ^ + e3 sin cos — y- -f Ml / mc v exp \2kT\ Then ( 2kTm 2) \ 2kT Av ^fAum^-Au, Using me v2 mc E M rn-x 2kT rn.. k T one obtains the following expression for the range: Avi Avj Aui Auj, ,c2kT {Av}=^fdQ~uAu, Nez2e4 7i lnil m, dl/ cLr (d-Q = sin 0 d0 dr/>) 2 m„ c1 Erf (Vx) 2e~x' 16 a In K {Avi Avj} = y— j f dD a Aut Auj. kT X y7i x 3 In A Erf (\'x) ( 1 -i------J --- -- Substituting for the cross-section (13) one has 2x) yn x , . ^ m,. 16are4 3 me me _E0 = ------~---—a'In AC, ma 3 mg~ c£ 2 ’ ma k T 140

~~542 P. S. Ray and H. Hora • Thermalisation of Energetic Charged Particles~~

~~10° I0’~~

Plasma Temperature Electron Volts Plasma Temperature in Electron Volts Fig. 3. Range of 3.6 MeV alpha in DT plasma of solid state electron density. The dashed line shows values according to Fig. 5. Range of alpha with initial energy 2.894 MeV in Winterberg formula. Hydrogen-Boron plasma of solid state electron density.

io‘ gen-boron and deuterium-tritium plasmas. We have also shown in Fig. 3 the deviation from our results to1 of the Winterberg J values usually used in literature. E The above result differs from the Winterberg values V to3 at the lower and higher temperature ends. One notes in Fig. 4 that for the case of 14.7 MeV 10 protons released by fusion in DT plasma the range first decreases with increasing temperature and sub 10° sequently increases. This behaviour is not shared 10° io’ IO3 105 IO1 10s 10s to3 by the alphas. The numerical value of the range is Plasma Temperature Electron Volts thus very susceptible to the ratio of the electron Fig. 4. Range of proton with initial energy 14.7 MeV in DT mass to the heavy particle mass as well as the initial plasma of solid state electron density. heavy particle energy. Using the cut-off of the w,nax/ a>mjn as above, we can note that the numerical in if kT is in electron volts one has for cgs units fluence of the photon emissions during scattering is not substantial although this aspect has to be taken ma 10* x (6.9 xlO6) into account for a correct theory. This is mainly me N e z- 2 7i\n A [ ’ because of the enormous mass difference of the elec f cLr trons and the heavy ions. However, for the case of J io6 Erf (Vx) 2 e~x external relativistic electron beam heating this could he of importance. kT X Vji X One should furthermore note that the larger the 16 a In K f .- / 1 estimate for log (a)max/(Oinin) is chosen, the smaller Yti X + T*uTa \™(Vx) [~1 + J-a the computed range would become. This is also where feasible since emission of very soft photons during (rn±___x (kT \3'2 \ In k — In scattering should make this factor large. However, \mc V'4 7i Nc \ e- ) / in the absence of experimental data for the range it is difficult to form any opinion on this aspect. 4. Numerical Results In the adjoining figures we show the numerical Conclusion range values as a function of plasma temperature It is thus seen that usually accepted formula for solid state electron density of alphas in hydro R~T^2 giving range as a function of plasma tern- P. S. Hay and H. Hora • Thermalisation of Energetic Charged Particles 513 mature for alphas in DT is not in general valid for the range if one chooses a large enough value for her cases. The emission of photons during electro- log (wui;lx/o)min) where and

D. F. Diichs and D. Pfirsch, Proc. of Conf. on Plasma I K. A. Brueckner and S. Jorna, Rev. Mod. Phys. 46, 325 Phvsics and Controlled Nuclear Fusion, Vol. 1, Tokyo [1974]. 1974, IAEA, p. 669. 8 K. A. Brueckner, II. Brysk, and R. S. Janda, J. Plasma J. R. Kerns et ah, Bull. Amer. Phys. Soc. 17, 690 [1972]. Physics 11, 403 [1974]. E. Bagge, 13th Int. Cosmic Ray Conf. Denver, Aug 1973, 8 R. P. Feynman, Quantum Electrodynamics, Benjamin, The Origin of Cosmic Radiation (Thiemig, Munich) 1968. New York 1962, p. 150. E. Bagge and H. Hora, Atomkernenergie 24, 143 [1971]. 10 H. Tsuji et al, Nuclear Fusion 16, 287 [1976]. F. "Winterberg, Desert Research Institute Reprint Series, II M. N. Rosenbluth et ah, Phys. Rev. 107, 1 [1957]. Nn 61 ATnrrh 1969. 11 T. Biorken and S. Drell. Relativistic Quantum Mechanics. 45ERJf'ITEiFACT,0N AND RELATED plasma phenomena, VOL. u 142 Edited by: H. J. Schwarz and M. Hora Book available from: Plenum Publljhlng Corporation 227 West 1 7th Street, Now York, New York 10011

~~CORRECTED PENETRATION LENGTH OF ALPHAS FOR REHEAT~~

CALCULATIONS*

~~P. S. Ray and H. Mora~~

~~Dept, of Theoretical Physics, The University of New South Wales,~~

~~Kensington, 1NISW Australia.~~

~~Abstract~~

The penetration length R of alphas, protons or other particles from nuclear reactions in high density plasmas is essential for the reaction yields in laser compressed plasmas, high energy particle beam interaction with dense plasmas and other reactions. The often used formula It ~ T3/2 gets a remarkable correction for T between 101 and 103 eV, if the thermalization is described by a Fokker-Planck equation taking into account the elastic as well as inelastic scattering. The results derived from a collective interaction similar to Bag.ge's generalization of the Eethc- Bloch-formula, differ again from the mentioned former values despite the corrections discussed. The values of the collective model may have the highest probability as they agreed for fast electrons with the only direct measurements available.

~~I INTRODUCTION~~

In high temperature plasmas for thermonuclear reactions, the penetration depths of the charged nuclear reaction products are of essential importance for the energy balance of the plasma. This self-generated heating of the plasma is called reheat and plays an important role e.g. for laser compressed plasmas1 to reach self- ignition2 or laser induced self-sustained nuclear combustion fronts3.

The usual method for calculating the penetration depth of protons, deuterons or helium nuclei of IvIeV initial energy in plasmas uses Coulomb scattering for evaluations in a Boltzmann or in a Fokker-Planck equation4 where the low angle scattering is to be considered. The numerical evaluation of these mechanisms lead to a penetration depth R in dependence on the temperature T for the McV *Presented at the Fourth International Workshop Conference on “Laser Interaction and Related Plasma Phenomena” at Rensselaer Polytechnic Institute, Troy, New York, 8-12 November, 1978. 143

~~alphas of a DT reaction in plasma of solid state density5~~

~~R ~ T3/2 (1)~~

where R is in cms and T in eV. This result was used for the numerical calculations for laser compression and ignition2 as the next available relation though the authors involved were aware that the basic theoretical assumptions may be insufficient. The problem of other nuclear reactions taking place with these energetic particles has been expressed in some general considerations by some authors6.

\ The problem of reheat is also of essential importance for the design of magnetically confined toroidal configurations, as Tokamaks, where a detailed calculat ion of the temporal development of the distribution function of the generated alphas from the Fokker-Planck equation illustrate the processes involved7. This development is very complex as discussed by some8 — but it is difficult to draw any definite conclusion from this regarding the penetration depth.

Generally speaking, there are basic problems of the Coulomb scattering Involved apart from collective effects of plasmas, which even may be dominant at high densities. But alone the Coulomb processes are connected with the complex question of wide angle and low angle scattering. The wide angle scattering can be assumed as sufficient to calculate collision cross-sections of electrons or other mean free paths at thermal equilibrium9 in full agreement with the knowledge of resistivity in plasmas (cr metals). This agreement holds even for very high frequency resistivity, as e.g. in the calculation of the optical constants based on Spitzer’s assumption for the collision1»’ 0 as can be proved by the agreement with the quantum mechanical calcul ations of the inverse Bremstrahlung1 . This agreement even holds for the nonlinear deviations of the optical constants for very-high laser intensities 1,12’lJ’14. Deviations are within factors of the order ±5, given by the ratio of the Coulomb logarithm of the plasma case to the electrodynamic interaction logarithm of the quantum mechanical case. It has to be noted that for superdense plasma no quantum theory has been presented and the plasma values are based on the collision processes, on the one side and on a phase mixing treatment of the Vlasov equation13 on the other, both in some agreement.

While for the cases of wide angle scattering, the inelasticity of the Coulomb scattering may not be essential,the slowing down process of energetic heavy particles in a plasma has to include the inelastic contribution for a theoretically satisfactory point of view. The low angle scattering may not be the most effective slowing down process, as can be seen very globally; the penetration depths of particles at temperat ures above 10 keV can be 1000 times larger from Eq. (1), than the penetration in cold, unionised materials, as known from the Bethe-Bloch formula16’17. This difference did not disturb much, because there were as good as no measurements available to see the differences.

One measurement with relativistic 2 MeV electrons18, however, showed a discrepancy generally. The currents of 10s Amperes of electron beams hitting solid CD2 - targets transformed these immediately into plasma and it was very remarkable that the penetration depths dropped by a factor of about 20 below the value of union- 144

iscd cold solid targets. All artificial attempts to explain the measured low penetration of the electrons by the usual models of long lengths with turning of paths could neither arrive at the lew values nor could such models1 9, be considered as a realistic mechanism.

One can formulate a basically different description of the penetration depth of alphas by including ‘'collective” effects, at least for plasma densities around and above the solid state. The way to this model was given by Bagge by a generalis ation of the Bethe-Bloch formula20 where the averaged binding energy of 14.6 eV of the atoms had to be substituted by the electrostatic energy c2 /Xjj (\p) Debye length) which we are now able to explain as a plasma binding energy. Bagge calculated by this way drastic changes of the behaviour of cosmic ray electrons in surprising consistency with observations2 0. It was further possible21 to calculate the measured reduced penetration of the relativistic electrons in high density plasmas directly without any complicated assumptions.

This result encouraged us to study the case of alphas where, however, we did not use the Bcthe-Bloch formula but the usual concept of Fermi22. We arrived at also in this case penetration depths for high temperature plasma of solid state density values less by a factor 10 to 20 below the range of alphas in cold solids2 3’24. Apart from this treatment, we checked whether the concept of the low angle scattering could be altered by including the properties of inelasticity by including the emitted photons. The suppression of the infra-red catastrophe is possible again by using a “Bagge frequency” in analogy to the mentioned modification of the Bethe-Bloch formula. We also tried a suggestion of Pauli25, but the penetration depth obtained were all very different. We are now comparing the different models and will draw conclusions for the possible numerical results.

~~II THE FOKKER-PLANCK FORMALISM~~

The Fokker-Planck (F.P.) formalism describes statistically the interaction of a system of different species of particles. Let subscripts s, t, . . . denote the physical entities corresponding to the species e.g. fs (r, v, t) the distribution function for species s in a collection. Its evolution is governed by the F.P. equation

~~3f 3f F • 3f CO (-l)n ~ (il) , tS r; \ .—E + v. + -A _s ZZ D' ' (a , x f ) (2) 3t l 3x. m 3v. n l (n) s l sitn=l~~

~~Here the sum t extends over all species, Fj the component of the external force (i = 1, 2, 3) and we use the notation~~

~~(n) (each i = 1,2,3) 3v. 3v. ... 3v. ii 12 i (3) ts ts , a , x = a (v. v. ...v. v (n) ii i2 iR)~~

~~The a ’$ are the F.P. coefficients. This form of the equation was used by Kramers26. 145~~

We note that there arc two different ways of regarding the right side of Ecj. (2). In the intuitive picture it represents an approximation to the Boltzmann collision integral. This view is taken by Roscnbluth et. al.4 who have calculated the F. P. coefficients on the basis of Rutherford scattering. In the other point of view one regards (2) as a description of a stochastic process described by the Chapman-Kolmogorov equation. The right side is then simply a Taylor's expansion. We note that there is no assumption for thermodynamic equilibrium for the validity of (2). We shall denote the heavy particle by suffix H and the electrons by e. Because of the enormous mass difference of the electrons and ions one can neglect the ionic effects for processes of energy exchanges. Let i^(v) denote a function of velocity v only. Its average \j/ with respect to the distribution f is then defined as

~~^ = /.li'fdv = i /i(ifdv (4) / fdv~~

~~(dv = dv i dv 2 dv 3) where n = n(£ t) = / fdv is the local density.~~

~~Multiplying (2) by and by integration one obtains~~

~~(5)~~

where averaging is taken with respect to the heavy57 particle* distribution function. Here aJ,^) denotes the F. P. coefficients of order m describing interaction of electrons and the heavy7 particle. By partial integration one can write the above as

~~He __ (m) , r: (m)D *fHdv 146~~

~~In the absence of externa) force one can then obtain~~

~~(H) 3_ ~ 1 w He _ (m) . ) Z —f n ct , . D i\> 31 m=1 ml II (m) r~~

~~(6)~~

~~He 3mib where a^e ib a ----v. ) m y i 3 v. 3 v. ...o v. m. li i2 i s~~

~~Now setting 0(v) = Vfc one gets from (6) since all derivatives of \Jj higher than second order are zero 1~~

~~1 _3 2 31~~

n m (a. v. ^ i b.. 6.. ® H H i l 2 13 13 (7) with aj =* ci(vj) and bjj = a(vj,vj) are the first two F. P. coefficients. Eq. (7) is an exact relation. To pick a particular particle with coordinate and velocity (r pj, vp{) we set

~~fH = <5 (r - ?H(t)) 6 (v - vH(t)) giving nR = 6 (r - r ^ (t) )~~

~~\p(v) = (Vjj)~~

~~One then obtains after integration with respect to r and v~~

~~(Z L • VTT . ( ) 1 H, 1 + 11 bii> 8 i 1~~

This is the fundamental relation expressing the energy-loss by the F. P. coefficients. Rosenbluth et.al.4 have calculated the latter when only elastic scattering by Coulomb interaction is taken into account. This result is mostly utilised in plasma physics. One can also use the time correlation technique2 7 to calculate these. However, both results arc the same. These give for the F. P. coefficients the following values

~~ai = r h('^ 147~~

~~b ij (9) J~~

~~where 1 e~~

~~g (v) = / d v* fe(v1)~~

~~Here Z is the charge of the heavy particle and A represents the Coulomb logarithm~~

~~- where Ne is the electron density and kT the plasma temperature.~~

~~We note that the origin of this term is the divergence at 0 = 0, where 0 is the scattering angle, of the classical cross-section~~

~~(10)~~

with mr denoting the reduced mass and u the relative velocity. To get over this, one is forced to introduce some cut-off 0 = 0mjn. To estimate one assumes on an intuitive basis that small angle scattering corresponds to large impact parameter and as such one can employ the Debye shielding in plasma. However, this matter has always been of weak foundation in plasma physics. In the following we derive the expression for the range basing on this formalism. For the electron distribution function fe(v) we use the Maxwell-Boitzmann distribution

~~fe(v)~~

~~One then obtains i -yv’2 1 where I = /dv e 1 |v - v'l~~

~~\~\>~ 3/2 1 y II - Erf (y 2 v) v and Erf ;y e-fc2 dt the usual error function. i'> ■ /f J 0~~

~~Similarly E = T l v. a. 3v. h<"> 1~~

~~= r (i + V) n (*•) 3/2 81 in ' e 'tt V 3v e Utilising mjj » mc one obtains~~

m v dE e H 4ttg4 Z2 i a tvt ,2 / e 2kT 1 / e = ------InA N ( e ViT 7 2k¥ e “ v>> (11) Nov/ the range is defined as the distance travelled until thermal energy E^ = 3/2 kT is reached. Thus if E0 is the inital energy of the heavy particle the range R is given by

~~fEth, K dE Eo vdE J E j E ^-E- 0 cix th dt~~

~~X2 x3dx ~ (12) (—)2 f (x) e e2 N *lnA*iTZ2 Xi~~

~~i 9 - y: where f (x) = Erf (x) - — xe 149~~

As kT increases x2 ■> x, so that the integral in (12) decreases. However, the factor (kT)2 increases much faster giving a range which increases with temperature. The rum erical values of this is discussed below.

~~Ill PHOTON EMISSION IN SCATTERING~~

The above result is based solely on the elastic scattering. However, this is not entirely physically correct. As has been very explicitly emphasized by Feynman28 in the actual process there is always some photon emissions and this should be taken into account. This is represented by the following Feynman diagrams (Fig. 1)

~~Fig. 1 Feynman diagram showing photon emission in electron-scattering in external field represented by cross. There are two such diagrams contributing to the process.~~

The cross section for this process was calculated by Schwinger29 in a relativistic way for the first time. It can be calculated in a simpler way by Feynman rules30. In the non-relativistic limit this is given by ,do. 2a. elk 4_ p 2 sin2 0/2 (13) ^a^ ph {an}c TT k 3 ^ where cc is the fine-structure constant (oc = 1/137) and (i ~ u/c. The factor dk/k means that the photon of momentum between k and k + dk is emitted. With respect to the photon frequence to : dk/k = dco/to. To find the contributing cross-section one has to integrate over all frequencies emitted say from comjn to comax giving

~~^dq^ _ 2 a Z2e4 ^ ^max ^ _____ 1_____ (14) dft ph 3t\c2 m2 wmin u2sin20/2 e '~~

One notes that if ojmin + 0 then da/diZ •> °o which is the famous “infra-red” divergence of quantum electrodynamics. Similarly the divergence associated with cornax -> °° is the ultra violet catastrophe. One has to use again some cut-off values to get over this difficulty. For cj>m}n we may choose the “Bagge frequency” cog defined as e2/Xg) = hcou. Bagge31 first utilised this for the penetration of electrons in cosmic rays as mentioned above. For com2X we utilise the instantaneous energy of the heavy particle: E = ficomax. One thereby obtains 2q Z2e4 (—) In K ------(15) vdfT ph 3fre2 rn2 u2 sin20/2 where kT K = 4ttN e2 e To incorporate this process one may use the sum of the two differential cross-sections; da __ ,da ,da > aa ldfrph [an}c (16)

~~This will lead to additional terms for the F.P. coefficients a;s and b|js. We work out these in details now'. We have~~

~~a. = / dv ’ f (V-) {Avi}iIe 1 e~~

~~b.. = / dv’ f (v1) {AVi Av.}He 13 e~~

~~Here (Av) denotes the change in velocity per unit time due to collision. Suppose v and V denote the velocity of the heavy particle and the electron before collision. 151~~

~~Then their relative velocity u and centre of motion velocity V are given by: u = v - V ' •> V = V + M u r* mH ->• V* = V - -M U M - me + “h -> MV mrT v + m Jrl 1~~

~~Now the change in u by collision, denoted Au, can be obtained by consider ing the figure (Fig. 2)~~

~~Fig. J2 C;, e2 and e3 local axes system, 0 scattering angle, Ail change in u by scattering.~~

~~|Au| = 2u sin 0/2~~

~~Au = |Au|[-si sin0/2 + (e2 cos$ + e3 sinc^) cos0/2 ]~~

~~m m Then o -y p -* Av = —Mrr Au - m— Au~~

~~m 2 (—) Au. Au . mn 1 J One obtains~~

~~{Av} = ~ f dft ~ u Au~~

~~in 2 _e {AVj. Avj} “ s dn in u Aui Auj lH~~

Here df2 = sin 0 d0d$ and if one substitutes the new cross-section (15) one finds that it is not necessary to introduce a small angle cut-off 0m\n. Thus the Coulomb logarithm does not appear in this case. However, there is another logarithmic term due to the divergences of quantum electrodynamics. After simplifying the algebra one obtains

~~{ A u } = - —*e ------16aZ2e4 ln , Te_ l 1-H 3m2 c2 e~~

~~a - / dv' f (v1) {Av}~~

~~T<\ -L r ry 2 U . ~y ~y , = _ 16,3.2 e ln R /d+,f (+, ) ^vj mH 3mr>2c2 e Iv-v* Since V-V —3 I -vy - -vy , we can write I *> ~y , I V--V ' 3v~~

~~e 16aZ 2^4e a ln K N (-)Y.3/2 £l e 7T -> H 3m 2 c 2 3v e where -too 2 ->- -r , —/ CO chP e"YV v-v~~

~~V(v) Y3/2~~

~~Y(v) (e YV - X)+/ir Erf (/yv) (v2 + -=i) *• Y /Y 153~~

~~Thus 'e 16aZ?e4 a • v = —------In K (^-) V2N [I - 71 V (v)] H 3m2c2 3/2 e Y Similarly \ z bi± = \ £ * dv' fe(v') {Av± Av±} i i~~

~~(—) I£“Z_E_ ln K N C1-)*/*! ™H 3m 2c2 f. 71 s e • • 1 “F- giving 2 Z << a»v. i~~

~~For the rate of energy loss one therefore obtains . o2 4 to /m m v2 dE 47tZ e i » 1 r 2 / e , e x dt ~ in Ne lnA vC7^/ 2kT V exp ( 2kT^~~

~~- Erf ( 2kT v)]~~

~~+ ■l0.aZ.--cl in K N - - [/? Erf ( V) 3/rrm c2 e v y 2kT m v / m m v2 X 2kT~ + l) 2kT v exp(~ 2kT~^~~

~~One can then write the following expression for range 2 l7 me kT x2 _H e______dx~~

~~le N Z2e4TrlnA e Xi 2mec rErf(/x) _ 2e + kT ' x " /^~~

~~il2- x * {Erf (/£) (-1 + -=|) - —} (17) 3 lnA 2x~~

~~e Q and with xi x2 kT If kT is in electron volts one obtains in cgs units~~

~~R = ^ 10 6x(6.94x106 ) 2 (kT)~~

~~me Z2 N 2it InA 10GrErf(/x) 2exp (-x) Xl kT [—x------7=------•] / 7fX~~

~~exp (~x) y /ttx~~

~~(18)~~

~~H x kT 3 / 2 where In K = In ( (—) ) m / 4ttN e2~~

~~In quantum electrodynamics one has another conjecture of Pauli2 5 which seems to be supported by Landau31 to use the cut-off~~

~~max _ 1 137 log 2a 2 min~~

~~In this case InK = 68.5. The numerical results are discussed below.~~

~~IV COLLECTIVE INTERACTION MODEL~~

As discussed above the usual F.P. formalism is plagued with the problem of cut-off - be it the small angle scattering or the divergences of quantum electro dynamics. And the range will depend very significantly on this. However, one can consider this problem under the aspect of a “collective” effect as formulated by Gasiorowicz et.al.32. One considers here the interaction in basically two parts i.e. one with the immediate neighbourhood where two body type collisions are important and the other with the rest of the plasma. The presence of the energetic particle causes a perturbation of the distribution function of the plasma electrons, some sort of a “polarisation”, which then acts back on the particle. Now one can divide the total interaction in two parts - one with the immediate neighbourhood and other with the rest of the plasma. To define the former, one may use the Debye radius Xj) as a cut-off factor within which binary type collisions are important. The component of the force in the direction of motion Fv due to the polarisation effect as calculated by Gasiorowicz et.al.3 2 is given by 2e2 t ^ 9 x r+1 log (k dp p J(p, v) (19) max A2) '-I 155

where the function J is defined in their paper. Here kmax is a cut-off which is an inverse of a length denoting dynamic correlations of two particles when they are close together. If one uses kmax ~ 1/An one sr“os that Fv ~ 0. One can use a two- body picture of the energy transferred to electrons within the Debye sphere as given by Fermi . An electron e situated as shown in figure at a distance r with respect to the heavy particle Ze will in general experience a nett force perpendicular to the direction of motion of Ze. If b denotes the impact parameter the average force is

~~Ze cos G = Ze2 — >-2~~

~~Fig. 3 Ze heavy particle, e electron at an impact parameter b from it.~~

~~If it acts for time At the gain in momentum by the electron is then~~

~~Zez b Ap Ze2 - At~~

~~where v is the velocity of Ze (At/Ax = 1/v) and Ax the distance travelled by it in At. The total momentum Ap gained by the electron is then~~

~~-foo Ze2 b dx (Ap) total~~

~~Ze2 2 bv 156~~

~~If AE denote the total energy gain~~

~~(Ap) 2 total AE 2Z2 e4 2 m b2v2m e~~

~~Now the number of electrons situated per path Ax of the heav>r particle between impact parameter range b and b + db is 2tt Neb db Ax. Thus the total energy loss to the electrons per path Ax is~~

~~f max 4iTZ2e4 -AE = N bdb Ax e jk b2v2m J b . e mm b dE 4ttZ 2 e4 .. max ------log dx b tn 2 min~~

~~If one uses for minimum and maximum values of the impact parameter the following values~~

~~Ze min m v' e which is the classical value and bmax Xg) = •v/kt/47rNee2 the Debye radius as mentioned above one obtains~~

~~kT m 2v4 dE _ 0 Z4e4 XT , , -r— ~ 2tt ------N log ( dx 2 e 3 m v 4rrN e6 Z2 e e which is similar to the liethe-Bloch formula. Now v2 = 2E/mpj so that one can write~~

~~raH Z 2 e4 kT m 2 E2 —---- N log ( m E e ^ it N e Z2 mH2 e5 rx et Using the integral logarithm jrp (x) dt one can express the range R as~~

~~? m.r dE e H Ei (log (x E0 2)) ( ) dE 2kT m 20 where where 10 3 +~~

~~is One The solid Fig.~~

~~RANGE IN CMS in~~

~~electron 4~~

~~criterion range state~~

~~density. of~~

~~volts~~

~~to 2.89 kT~~

~~and~~

~~satisfied >> MeV~~

~~N tt c 2.413~~

~~alpha is in~~

~~in is H kT 2~~

~~that cm~~

~~Z in m~~

~~x 2~~

~~b 3 IIB(ll) max N 10 2~~

“

~~» e 25~~

~~1 plasma~~

~~b N m i n~~

~~which of~~

~~equivalent~~

~~to ( 21 ) TEMP IN EV~~

~~Fig. 5 The range of 3.6 MeV alpha in DT plasma of solid state density based on Fokker-Planck formalism. The dashed line corresponds to Winterberg values.5~~

~~TEMP IN EV~~

~~Fig. 6 The range of 14.7 MeV protons in DT-plasma of solid state density calculated on the basis of Fokker-Planck formalism. 159~~

~~INPUT LASER ENERGY IN JOULES~~

~~ION DENSITY ~ 9-4 - 1024 cm'3 V0 INITIAL PLASMA VOLUME~~

~~Fig. 7 Fusion yield for II1 + B1 1 •> 3He4 + S.6S2 MeV reaction. Lower graphs correspond to without alpha reheat, upper ones with alpha reheat.~~

~~V. NUMERICAL RESULTS~~

The usual F.P. calculation together with photon emission correction which is first discussed does not alter the numerical values significantly although this theoretical aspect one should always consider. The Pauli cut-off shows a little lesser range at high temperatures. One remarkable aspect is that the proton range (14.7 MeV) for DT plasma of solid state density first decreases and then increases. Thus the physical character of the range when calculated by the F. P. formalism is susceptible to the parameters as the mass, initial energy and charge of the heavy particle. In the above figures numerical values for certain cases are shown. The effect of the absorpt - ion of alphas oroduced by fusion reaction

~~II1 +B1 1 * 3 He4 + 8.682 MeV on the fusion yield has also been calculated and shown in Fig. 7.~~

~~VI SYNOPSIS~~

The above ways of calculating give physically different results as can be seen from the fact that the range, in general, increases with the plasma temperature for the F. P. formalism whereas it decreases for die collective type interaction. In the absence of experimental observation it is difficult to form an exact opinion on the correct behaviour on this matter. However, there seems to be good ground to suspect that the collective approach calculation should correspond more to reality. Firstly we emphasize that the divergence of Coulomb cross-section for 0 = 0, which has been cut-off by 0 = 0mjn and thus forced to be finite, would have implied that the F. P. coefficients arc infinite and thus that dE/dx also infinite. This would have meant that this range is zero. Besides the usual scattering cross-section utilizes the classical elastic type collision which is least suited for the purpose of energy exchange. Different cross-sections available from quantum electrodynamics are also objection able due to the presence of divergences which would again imply zero range. The collective type approach gives the expression for the range close to the Bethe-Bloch formula which is very well in agreement with the nuclear physicists. Mere the neglect of interaction beyond the Debye sphere would mean that the range could be even shorter.

One important point to be noted is that for very low temperature regions one cannot apply the usual F. P. calculation since the Coulomb logarithm becomes negative for high density plasma. The restriction (21) is however quite within the range of plasma temperature.

~~VII CONCLUSION~~

It is conjectured here that for the calculation of range of energetic charged particles in plasma of high densities one should employ the collective type approach which gives ranges which decrease with the plasma temperature. In the absence of direct experimental evidence, it is difficult to establish this fact. The usual F. P. formalism at least with the known differential scattering cross-section may not be suited for this purpose. Since the energy yield in fusion reactions depends on this alpha particle heating of plasma which in turn depends on the range it might alter the usual values in this respect. Furthermore, certain pictures2 of nuclear ignition which depend on the range may not correspond to experiments then. REFERENCES

~~1. H. Hora, Laser Plasma and Nuclear Energy (Plenum, 1975).~~

~~2. K. A. Brueckner and S. Jorna, Rev. Mod. Phys. 46, 325 (1974).~~

~~3. J. L. Bobin, Laser Interaction and Related Plasma Phenomena, H. Schwarz and H. Hora, Eds. (Plenum, 1974) Vol. 3B, p.465.~~

~~4. M. N. Rosenbluth, et.al., Phys. Rev. 107,1 (1957).~~

~~5. M. S. Chu, D Thesis Columbia University (1971). F. Winterberg, Desert Research Institute Reprint Series No. 64, March (1969).~~

~~6. K. A. Brueckner et.al., J. Plasma Physics, 11, 403 (1974).~~

~~7. PI. Tsuji et.al., Nuclear Fusion, 16, 287 (1976).~~

~~8. D. F. Duchs and D. Pfirsch, Proc. Conf. Plasma Physics and Controlled Nuclear Fusion, Vcl. 1, p.669, Tokyo (1974) IAEA.~~

~~9. L. Spitzer, Physics of Fully Ionized Gases (Wiley, 1967).~~

~~10. H. Hora and H. Wilhelm, Nuclear Fusion, 10, 111 (1970).~~

~~11. S. Rand, Phys. Rev., 136B, 231 (1964).~~

~~12. T. P. Hughes and M. N. Florence, J. Phys. A (2), 1, 588 (1968).~~

~~13. H. Hora, Opto-Electronics, 2, 201 (1970).~~

~~14. B. Rabuel-Peyrissac, 6th Int. Q. E. Conference (Kyoto, Sept., 1970), Conf, Digest p.14.~~

~~15. J. Dawson and C. Oberman, Phys. Fluids, 5, 517 (1962).~~

~~16. PI. A. Bethe, Ann. d. Phys. 5, 325 (1930).~~

~~17. F. Bloch, Ann. d. Phys. 16, 2S5 (1933).~~

~~18. J. R. Kerns, et.al., Bull. Am Phys. Soc. 17, 690 (1972).~~

~~19. G. Yanaes, Ettore Ivlajorana Course Erice June (1973).~~

~~20. E. Bagge, 13th Int. Cosmic Ray Conf., Denver, Aug. (1973).~~

~~21. E. Bagge and H. Hora, Atomkernenergie, 24, 143 (1974).~~

~~22. E. Fermi, Nuclear Physics, p.28 (The University of Chicago Press, 1963). 23. P. S. P^ay and II. Ilora, Nuclear Fusion, 16, 535 (1976). 162~~

~~24. P. S. Ray and H. Iiora, Atomkernenergie, 28,155 (1976).~~

~~25. W. Pauli, Helv. Phys. Acta, Suppl IV, p.6S (1956).~~

~~26. H. Kramers, Physica, 7, 284 (1940).~~

~~27. W. B. Thomson and J. Hubbard, Rev. Mod. Phys. 32, 714 (1960).~~

~~28. R. P. Feynman, Quantum Electrodynamics, p.150, (Benjamin, 1962). \ 29. J. Schwinger, Phys. Rev., 76, 790 (1949).~~

~~30. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, p. 126 (McGraw- Hill, 1964).~~

~~31. L. 1). Landau, “Niels Bohr and the Development of Physics” (Pergamon-Press, London, 1955).~~

~~32. S. Gasiorowicz, et.al., Phys. Rev., 101, 922 (1956). 163~~

~~ADVANCED FUEL NUCLEAR REACTION FEASIBILITY using laser compression ii~~

~~R CASTILLO, II. IIORA, E. L KANE, V F. LAWRENCE, M B. NICHOLSON-FLORENCE, M M . NOVAK, P S. RAY, J. R SHEPANSKI, R L SUTHERLAND, A. I TSIVINSKY and H. A. WARD~~

~~Department of Theoretical Physics, University of New South Wales, Kensington. Sydney, Australia~~

Several problems are considered in order to study the properties of the nonlinear-force compression of plasmas by lasers used to reach conditions for advanced clean fuel nuclear reactions as well as to distinguish Irom gas-dynamic compression. The propagation of light in inhomogeneous media is based on a simpler, computationally economic programme. The Goos-Haenchen effect is used for discussions of wave propagation and a laser amplifier w-ithout superradiance designed. Corrections for ar-reheat are derived and a very short-range relativistic self-focusing discovered w'ilh relatively low thres holds. Entropy production and electron-radiation interaction are treated relativistically.

1. Iintroduction centrated within the last lOOps, its intensity must not exceed such values which result in collision Fallowing the preceding article1), there exists the times longer than the interaction times. These ab possibility of generating clean nuclear energy from solute limits restrict the neodymium glass laser in laseir compressed plasmas, if for example the reac tensities to") less than 10l6w/cm2 also if instabil tion H + MB = 3a is used. One necessary condition ities generate an anomalous effective collision fre is tlhe use of the nearly isentropic transfer of the quency higher than the nonlinear Coulomb colli laseir energy into kinetic energy of plasma2), which sion frequencies12). On the other hand, a power is olf such high efficiency that the advanced clean density of 10l9W/cm2 is necessary to compress fuel reaction becomes feasible at laser pulse en the plasma for nuclear reactions13). One way out ergises of less than 1 MJ. Compression based on may be the gasdynamic compression scheme of the gas-dynamic ablation3) results in efficiencies Afanasyev et al.14) in addition to the compression mucch lower than necessary to get feasible condi with the nonlinear force scheme1). tions. Though the MIGMA4) project, an alterna This paper discusses a series of several theoret tive for clean fusion, has reached an advanced le ical results which were performed to test the var vel,, there are some reasons still to proceed with ious aspects of the gas-dynamic and nonlinear for the nonlinear force compression method for laser ce compression scheme, where connections with fusiion by clean reactions. the problem of clean reaction of, for example, T'here are some experimental barriers against H + "B etc. are included; in addition radiation gascdynamic compression which have not been in problems are considered which themselves are si cluded in the present extensive numerical simula milar to those related to the laser produced pair- tion! for forecasting energy generation5). One of production215). thesse phenomena is the simple rule that the laser intemsity has to be less than 1014 W/cm2 for neo- 2. Propagation and reflection of waves dynnium glass lasers and 1012 W/cm2 for C02 lasers> for which, however, symmetric compres- For the action of the nonlinear force for com siom3ft) and reasonable fusion yields have been est pression it is of importance that the laser radiation ablished. At higher intensities, the phenomenon penetrates an inhomogeneous plasma with a min of Ifast ions has been observed78) which can be imum of reflection. It has been found for the case considered as an experimental proof of the action of linearly increasing electron densities that the of nonlinear forces9) similar to analogous cases generation of reflection is very strong and the re witlh microwaves10). sulting standing wave pushes electrons (plasma) One general theoretical aspect puts a further li- towards the nodes and causes dynamic absorp mittation to the intensity of the laser radiation for tion2) instead of thick fast moving blocks of plas gascdynamic ablation. While the main energy of ma. Though the example (see pp. 64-72 in ref. 2) the; highly sophisticated laser pulse is to be con still results in 23% transfer of laser energy into

~~ADVANCE;!) LULL FUSION FEASIBILITY STUDIES 164~~

~~28 K ( AS I I 1.1.0 Ct al~~

Reflectivities plasma with constant refractive index if enough 01 - steps (100 or 1000) are used. What is then called “internal reflection” is nothing else than the 00T/WVWWWWWWA/V a 12 3 4 5 distance yum WKB-like increase of the amplitude of the reflect 01 ■ ed mode which is singly produced at the interface of the Rayleigh-plasma to the homogeneous plas ma only18). The simplification of the described numerical treatment leads to a more precise solu tion of the wave equation in inhomogeneous med Fig. I. Reflectivity R of a set of vacuum - Rayleigh plasma (a = 104 cm1; A = 1.06 //m) - homogeneous plasma with con ia for general computer programmes. tinuous refractive index without collisions at various thick One open question is the result that the dielec nesses (I of the Rayleigh plasma. Case (a) is the exact solution, trically explained spreads of the minima of the case (b) is the approximation with 1000 steps of equidistant curves in fig. 1 are different for the exact case homogeneous plasma and case (c) with 1000 steps. (curve a) and for the stepwise approximation (curves b and c). It has to be noted that cases with net kinetic energy of a thick block of plasma, a small number of steps can differ from the results further suppression of reflection is necessary. in fig. 1 drastically. One example for low reflectivity is the use of a One further question of the propagation of rad density profile which results in an optical refrac iation in plasma is the momentum of the photons. tive index depending on the depth x In agreement with the recoil in inhomogeneous 1 surfaces919) and with the transport of a wave-- (0C > 0), 0) I + OCX ’ packet20), the photon momentum is

corresponding to an electron density nc for a col p = — 4 (1 - Wp/2u)2). (5) lisionless plasma of c n ne = ticcfl - 1/(1 +a.x)2] , (2) Peierls21) found a similar expression for transpar where //ct is the critical density (cut-off) at which ent solids first, modified, however, such that a the plasma frequency cop is equal to the optical basic difference exists compared with plasma. The frequency co. The case of eq. (1) has been first dis basic problem of the Abraham or Minkowski de cussed by Rayleigh (Rayleigh profile) and has ele scription has been discussed by Dewar22) and some mentary solutions of the wave equation of the aspects of the problem of the brief arresting of en type16) ergy with electrons during its exchange with the plasma electrons in connection with the Fizeau ef Ey(x) = (1 +ax)! exp ± i In (1 -fax) fect have been considered by Shepanski23). The switching-on and switching-off process of (3) the light when penetrating a plasma gives a van ia ishing net transversal motion20-24) also under relat Hz(x) (1 -fax)-* x 2w/z0_ ivistic conditions in difference to other authors25). The Goos-Haenchen effect (side shift of a wave at total reflection)26) is not only of fundamental in x exp + i In(1 -fax) (4) terest with respect to the correct use of quantities of phase or intensity, but also e.g., for the codes Reflection occurs at the interface between a Ray in laser produced plasmas. An important applica leigh medium and a homogeneous medium only, tion is in the penetration of radiation around a and not within the Rayleigh medium. The more spherical target within the skin depth, as has been general result of Osterberg17) on vanishing “inter pointed out for laser produced plasmas27) to ex nal reflection” gave rise to several controversies. plain some early experiments28). The inclusion into The solution to the problem can be seen from the numerical codes needs a clear analysis of the use calculation of a Rayleigh-like plasma of various of phases or intensity, which has been discussed, thickness between homogeneous plasma (fig. 1). for example, by Renard29). The exact solution of the reflectivity agrees in One consequence of these calculations is imme magnitude with the approximation of steps of diately connected with laser fusion, namely for la- 165 I AS! I< ( () M I* l< I S S I () N II 29

Rp lem is then, what mean free path or what pene ooooooooooooo tration depth R has to be used for the nuclei? The _m2 slowing down (equilibration) of fast ions in plasma is an old problem in plasma3*) and nuclear’7) phy ...______b, sics. One application for rclativistically fast elec B o q r> 000000 00.0 0 trons was made possible by Bagge's modification of the Bethe-Bloch formula of solids for plasma ”Vp R, and the derivation of measured penetration depth Fijg 2. Laser amplifier without supcrrailiancc The laser beam of relativistic electrons in plasma311). is incident at an angle a on a medium G adjacent to an opt ically inverted medium M so that x>o(c (crc critical angle of to- The application of the similar methods for tall rellection). The reflectivity is larger than one-14) and has alphas led to the penetration depth39) in plasma of specific maxima33). electron density /;e and temperature T

2 setr technology. Lasers always emit superradiation R = 2 — Ei[ln(x£S):. (6) k I mt ini addition to the desired giant pulses. This super- ratdiance had to be suppressed by 10 6 of the main where the function Ei(.v) is the integral-logarithm. la:ser power, before the first convincing fusion As a result, the length R can differ by more than neutrons were generated with neodymium glass a factor of ten from the measured lengths for sol latsers30). The same happened with C02 lasers31). ids of the same density. Fig. 3 shows the result Only the sufficient suppression of the superradi for solid state density of HB. The general values ance led to neutrons. For iodine lasers it is a of density are used in the computer codes; further nnuch more difficult problem if large cross sections theoretical work is undertaken to compare the dif otf the beam are used32)- One way to reach laser fering plasma theories. The correlation with the amplification without superradiance is described in depths in tokamaks, as determined by Diichs and fi)g. 2. Pfirsch40), is close though different models have 3.. Correction of the a-reheat been used. For the calculation of the cr-reheat in the codes In laser compressed nuclear reactions, the heat- the reaction gain G is given by: inig of the plasma by the reaction products, pro tons or alphas, is of importance and has been used KU) Mi[K(/)]' ini several detailed numerical codes35). One prob d.vdyd: (7) A Oc>,

~~where £, is the energy released per reaction, /;, is the ion density, A is 2 or 4 and~~

N T To + I A,Tt (Ko IK)2- (S> V — I PLASMA TEMPERATURE kT (electron-Volts) where Rv arc the radii at each time step of num Frig. 3. Penetration depth R of a-parliclcs from the 1111B rcac- tiion in plasma of solid state density of HB as function of the erical integration during which an increase of the ttemperature39). temperature A, 7g by reheat occurs.

~~I. ADVANCED FUEL FUSION FEASIBILITY STUDIES 16 6~~

~~30 l< ( A S I I 1.1.0 et ill~~

4. Relativistic self-focusing ergy multiplying factor for N = 1.0 and Z = 1; and for T — 1000 eV: jV = 1.0 and jV == 10.0, both for The self-focusing resulting from relativistic par Z= 1.0. ticle mass and energy influences becomes import N values > unity produce the cusp-like distribu ant for pre-focused laser beams of power greater tions with the extremely small minimum self-fo than 10lllW 4142)- Previous non-temperature de cusing lengths, the minimum allowable intensities pendent calculations given in ref. 41 show the deeply found in the relativistic regime with N$> 1. self-focusing length as a function of beam inten For /V - 1 the influence of temperature is minimal sity for nc/n™ ratios of 10 10 2, and 10 up to 100 eV; however, an increase of plasma where nc is the electron number density and temperature to 1000 eV yields a two order of /;N’K is the non-relativistic cut-off density of the magnitude decrease in minimum intensity. It is Nd glass laser (1021 cm •’)■ The magnitude of /S[ is seen that for /V values around unity a small inversely proportional to the square root of this ra change in /V produces a large change in both tio for //C///^0R = /V with /V<0.7; furthermore, the magnitude and shape of the /SK(/) distribution for distributions are roughly symmetric and convex /< /kel; the distributions are mutually closer with downward with minimums near /RlL, a characteris />/rel• The influence of temperature, for a fixed tic relativistic intensity (relativistic threshold) of N, is negligible for N^> 1. value 3.66x 10IX W/cm2 for Nd glass. It is of interest to qualitatively estimate the in Further new calculations for N values around fluence of non-collisional electromagnetic field in unity and above exhibit extremely small /sf. values, duced charged particle collective motions by intro as well as limiting minimum intensity levels for ducing a multiplying factor F (F < 1.0) times non-singular /S( results. Employment of the com plex refractive index, the nonlinear Lorentz gas - (*T + £k,n) lerms >n the collision frequency expres sion. This factor describes eventually occurring Coulombic expression for the collision frequency, instabilities by an “effective absorption” given by and a relation between the local laser beam wave length within the plasma to the beam radius41) gives — ~ z = 1. T= OeV. F = 1 0 'si' _ I (\mi)\ + |/i(//2)|V — ~ Z= 1.2.5.10 ; T= 100eV,

(K)) with <70 the beam width between the half-irradi- ance maximum points of the radial laser intensity distribution and v/w the normalized collision fre quency. It is readily seen from eq. (9) that for I//(//2)| > |//(/)l the solution becomes singu lar and then imaginary at this “threshold” inten sity /s; this occurence was not found in the cal N=1 5 culations of ref. 41 but is crucial in the conditions illustrated in fig. 4. In particular, the collision fre (o< r

an “effective collision frequency” vcfr=v/P'2. thermodynamical force of diffusion Thie results for N — 1.0, T= 100 eV, and P — 0.01 l = grad // + B dvjdt, (13) are: shown in fig. 4. The minimum allowable in tensity is increased by about an order of magni where // is the chemical potential and B is propor tude but the curve does merge with the other tional to the density. N — 1.0 distributions. On the other hand considering the particular lFinally, for /V=1.0 and 1.5, and 7’=100eV, case of a set of particles, confined in a box, it was fig;. 4 shows the dependence on ion charge num- shown that it is possible to get the conservation beir for Z = 1,2,5, and 10. The Z-influence does laws for the number of particles and for the en nod seem to be as great as for temperature; differ- tropy from the conservation of the ener ernces are negligible in the TV — 1.5 situation. For gy-momentum tensor. These expressions are simi thee most part, the minimum value of /s,,/<70 for a lar to the formulae given by Landau and Lif- pairticular N is adequately determined in a shitz44) for the case of a fluid. They also agree with T — 10 eV, Z= 1 calculation; the threshold inten remarks by Tolnian48) about the role of the mass sity /s for a physically correct relativistic self-focus and energy, in a relativistic theory. ing solution is nevertheless strongly dependent on These two results were obtained in connection T and Z. with the formulation.of thermodynamics of mov 1 Physically, the above results indicate the rela- ing systems. The case of a body in motion in pres tivdstic self-focusing is suddenly initiated as /, is ence of an external field dependent on time still adhieved, perhaps explaining the fast ion produc remains to be studied. tion7-8) resulting from self-focusing induced elec The case of a stationary field has been already tron oscillation energies £osc of a few 105 eV.* This solved for dielectrics, and also for a conductor us leaids to a nonlinear force expansion9) with trans- ing the Onsager theory46). Schmutzer gave a solu latiional ion energies e, given by exact integration") tion using the Onsager theory and taking into ac of count the gravitational field47). e, =Ze„,, (II) The way in which the conditions of thermody namical equilibrium are changed by the presence wiith Z the ionic charge. The observed ion energies of a variable field must be studied more closely. forr Al,l+ of 2 MeV 7), or for W20+ of 2 MeV 8) are This is of particular importance since these condi thcen of the right order of magnitude. The energy tions are basic to the application of the hypothesis deposited by the laser into the volume of the self- of cellular equilibrium, the concept underlying the foccused light cone corresponds reasonably well to whole Onsager theory. For one special case, the thee total energy of the accelerated plasma with problem of variable fields has been solved, namely MceV ions. for the generation of mechanical forces in the me dium. The generated recoil due to the variation of 5. Entropy generation and radiation problems the intensity of radiation in a homogeneous plas For the calculations of the entropy generation in ma20) is the same as the recoil of a constant ra thte numerical codes, the basic derivation of the diation to an inhomogeneous plasma9- 19). equation of motion, of energy conservation and Further problems of electrons in high intensity emtropy generation has been studied. It was shown laser radiation were studied with respect to radia thiat a term proportional to the acceleration of tion losses in pair-production, the energy ex plaasma in the generalized thermodynamical force change, showing a defect compared to the case of foiund in the relativistic theory43), can be obtained Einstein48) for nonrelativistically moving mole forr a stationary system using the hypothesis of cules. Novak has used for these calculations the cedlular equilibrium and introducing “constant anharmonicity of the oscillators in the black-body conditions”, in the energy current. radiation. We can then write in general, the relation 6. Conclusions X93 = grad T + A dv/dt, (12) For the propagation of laser radiation in forr the thermodynamical force xq causing the heat inhomogeneous plasmas, a more computationally cujrrent, where T is the temperature, i> is the bary- economic code has been developed for exact solu ce:ntric velocity, and A is a constant depending tions of the wave equation. The Osterberg prob om the density, and the following relation for the lem of vanishing generation of local reflection has

~~I ADVANCED FULL FUSION FEASIBILITY STUDIES 168~~

~~32 K C AS I 1.0 et al.~~

been explained by the WKB-like change of the 9) II. llora, Phys. Fluids 12 (1969) 182. wave amplitudes. The phase or intensity descrip *°) A. Y. Wong and R. L. Stenzel, Phys. Rev. Lett. 34 (1975) 727. tion of waves was studied within the problem of n) II llora, Austr. J Phys. 29 (1976) 375. the Goos-Haenchen effect for application to the ,2) S. Rand, Phys. Rev. 136 (1964) 13231; T. P. Hughes and propagation of radiation in the surface of laser ir M. B Nicholson-Florence, J. Phys. A(2) 1 (1968) 588; H radiated pellets. One consequence of the llora, Opto-Flectronics 2 (1970) 201. Goos-i laenchen application is the design of a laser 13) J« Emmett, J. Nuckolls and L. Wood, Sci. Am. (June 1974) p. 24. amplifier with complete suppression of superradi ,4) Yu. V. Afanasyev, N. G. Basov, P. O. Volosevich, E. G. ance, which is highly necessary for laser compres Gamalii, O. N. Krokhin, S. P. Kurdyumov, E. I. Levanov, sion of plasmas. V. B. Rozanov, A. A. Samarskii and A. N. 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